In Crash 1, Crash 2: Cortex Strikes Back and Crash 3: Warped, there are multiple places where it's possible for Crash to make a jump that wasn't intended by Naughty Dog to be possible. The reason there are so many in these games appears to come from the fact that Naughty Dog made an attempt to place platforms and gems just out of reach of how far they believed Crash could jump, instead of playing it safe. This was probably to make the player think a jump was possible, and keep trying, for instance, to reach the platform at the end of the blue gem path in Hang Eight. As a result, with perfect timing on both a slide and the corresponding jump, and sometimes spinning at the end of the jump, many of these jumps are indeed possible, but many of them are very difficult for most players to perform repeatedly.
It's possible to get all the crates without the green gem. This is done by using the crate barrage after the first checkpoint as a way to get to the iron arrow crate that leads to the three crates hidden. This can be done with, or without Aku Aku invincibility, but having invulnerability makes it much easier to break the three hidden crates. The camera doesn't move upwards when the player is in the hidden area without the gem. This method is employed in 100% speedruns to prevent the need to backtrack to the level after receiving the green gem from much later on.
At the end of the level, the player needs the yellow gem to access an extended part of the level. The need for the yellow gem can be prevented, however. There are two ways of doing this.
The first, more common way, works as follows: Jump on the last turtle before the regular end of the level to flip it over and begin bouncing on its belly. When the proceeding spike pole is coming down or at its lowest point, bounce straight into it, and Crash can damage abuse above the screen. As one damage abuses, tap the up button two or three times. Crash will be stuck above the screen, standing on top of the camera's view. Then, simply walk to the left, over the exit portal, and once Crash is above the platform on the other side at the start of the yellow gem path, press down to fall from above the camera. It really is important how long the player holds the up button when damage abusing, holding too long will cause Crash to fall right back down again immediately. Two or three taps or a very brief holding period will suffice. Obviously, an Aku Aku mask will be required. This method is employed in 100% speedruns to prevent the need to backtrack to the level after receiving the yellow gem from much later on. It is a much more easy and consistent method than the second method discussed below.
The second method is done by jumping on the extreme edges of the warp portal, then jumping onto the platform after it, at the start of the yellow gem path. This is easier done on PAL versions of the game, due to Crash's jump height being bigger. In NTSC regions, the player will need to zigzag while in the air to make the jump.
It's possible to enter and leave the blue gem path with an extremely precise jump, without the blue gem.
Crash Bandicoot 2: Cortex Strikes Back
Turtle Woods and The Pits
In the secret underground area in Turtle Woods, it is possible to reach the two high platforms by glitched high-jumping from the backs of the large purple ostriches even after they've already stuck their heads in the ground.
Both Turtle Woods and The Pits contain deep pits where Crash is supposed to bounce off of a rat or spin the rats and bounce on a bouncy mushroom platform to get out. However, it is possible to avoid every single one of these pits by using the side of the level and slide jumping. The player can leave the pits by doing a glitch high-jump too (a body slam is needed in the NTSC regions due to jump height being smaller).
It is possible to reach the red gem directly from the main part of level, using the nearby arrow crate or seal. The easiest way is to spin while landing on the arrow crate. This will prevent Crash from bouncing. Then, slide back to left, jump quite late to ensure Crash doesn't bounce off the arrow crate, then spin to get extra height as part a glitched high-jump. Crash should be just a tiny bit below the red gem by this point. Finally, belly flop, if the red gem is low down in its bobbing animation, the belly flop's extra height will allow Crash to claim his prize. The intended way to get this gem is to access the secret warp room via Air Crash. Note that the player will still have to unlock the secret entrance from Air Crash for 100% completion, as each secret exit utilized adds 1% to the total. In Crash Bandicoot N. Sane Trilogy the red gem was repositioned higher up to prevent that same glitch from happening.
It is possible to reach the platform at the end of the blue gem path. There's nothing special on this platform, as it only contains two Venus Fly Traps. This platform only exists for decoration. The big center part of the platform isn't solid either.
It is possible to go back up to the starting warp point by using a glitch high jump. Retracting can make the game freeze for a half-second, which is probably the reason the player isn't intended to return.
Road to Ruin actually contains three such jumps. The first is near the beginning where it is possible to slide and glitch high-jump to the left over a set of outline crates to an area that was only meant to be accessible via the secret warp room. The second is in the middle of the level, where Crash can jump from the main part of the level to the final platform of the death route, where he can grab the special gem without going through the death route. The third is that it is possible to slide jump back to the main level from inside the bonus round. The camera doesn't move since the game thinks Crash is in the bonus round, and if Crash moves too far, he'll fall through the ground as it isn't solid.
In the secret area, the swarmer rat platforms can be skipped by doing a glitch high-jump out of them.
It's possible to enter the death route from the back side, by jumping on the sides of the pit.
It's possible to enter the green gem route without the green gem using a glitch high-jump.
In the secret area, the swarmer rat platforms can be skipped by doing a sliding high-jump out of them.
It's possible to get back to the section with the death route platform without using the sparkly tentaclebot unit, by using a glitch high-jump.
Naughty Dog intentionally designed the level so that it is possible to enter the death route from the back side, since there isn't a pit there, making it so that players who die on their way to the death route can still enter it. However, by jumping on the sides of the pit, it is actually possible to enter the death route from the front as well, without using the death route platform.
Crash Bandicoot 3: Warped
If Crash has the Death Tornado Spin, it is possible to get the yellow gem without going through the secret warp room. He must bounce off of a trampoline near the end of the stage in order to bounce off a Basic Crate above it, then immediately double jump and Death Tornado to the building on the left. If done correctly, he will be standing on the corner. Crash must then peform a sliding double jump followed by a Death Tornado towards the gem, and then move carefully to the left at the last possible moment to grab it. Regardless of whether Crash obtains the Yellow gem or not, he will immediately die upon touching the roof where it lies because is not considered solid from the main area of the level, and will therefore be treated as a pit.
If Crash has the death tornado spin, then it's possible to enter the purple gem Path without the purple gem. For this, Crash must slide, double jump, and death tornado spin over from the main level to the end of the purple gem route.
It's possible to attack Dingodile without waiting for an opening and without the double jump power. This is possible with just a slide jump, although it's difficult, however a glitched high-jump can also be used for easier results.
At the end of the yellow gem path, it's possible to jump over the second wall of iron crates. However, nothing is solid after it, so the player will fall down and die.
The green gem isn't needed in this level's gem path, as it can be jumped over with a careful glitch high death-tornado double jump or a neutral slide jump and death-tornado spin combo. The blue and green sections of the gem path can also be skipped by going over the roof using the pillar at the right at the beginning of a path.