Is it possible to get "stuck" in Pokemon?

I've legitimately gotten stuck on pokemon blue. I went squirtle at the start and had no pokemon to learn cut and spent all my money. There was no way to buy a pokeball to get a pokemon that could learn cut and I had no way to get to lavender town.

Then in the next major town you can't get back to that guy unless you have cut.

I found a place in Terminus Cave in X/Y that I'm pretty sure you could get stuck in. There is a ledge somewhere that'll dump you in a place where you need Rock Smash to get back out. If you don't have Rock Smash you can't break the rock in your way, and you can't walk back up the ledge, so unless you have a pokémon with Dig or a pokémon that you can teach Rock Smash in your party, you're gonna live there forever. I'm not even sure if you could trade with someone. I don't remember if party pokémon show up on the trading screen.

You can always find a Wild Metapod while having a pure Metapod in the team. They will only use Harden, until they can't anymore and boom, this is how you get stuck.

I've always assumed that you could get stuck in the original games above Cerulean. If you didnt have cut, or couldn't use it, and then got that one trainer that protected that item ball to move as far forward as possible, then went behind him and then saved and turned off the game. He would move back to where he was once the game was reset.

I got stuck once. In gen II I was on the ghost type gym and I had my eevee out and it only knew normal moves and my team was fainted so i couldn't switch and his gengar couldn't hit me.I should of saved before the battle

You can get in a never-ending loop with the Old Man (Japanese Red, Green, and Blue)

In the Japanese versions of Pokémon Red, Green, and Blue, you could get stuck in an infinite loop with the Old Man (Video):

1. Have a full party and every box on the PC full
2. Ask the Old Man to teach you how to catch Pokémon
3. The Old Man goes to throw a Pokéball at the Weedle
4. The game will say that you have a full box and can't catch any more Pokémon
5. Steps 3 and 4 repeat forever and the only way to get out of it is to reset your game

The reason for this is described at Bulbapedia

This is because the game thinks that the player is catching a Pokémon, even though it's actually the Old Man. The game produces a "temporary" Pokémon to hold, and if the catching demonstration ends, that caught Pokémon is deleted.

This was fixed in the Japanese version of Pokémon Yellow and all non-Japanese versions of the game.

You can get stuck in a battle with Lorelei in the Elite Four and not be able to Struggle your way out (all Generation 1 games)

To do this:

1. Have just a pure Fighting-type Pokémon that knows Rage in your party (such as Primape)
2. Deplete the PP of all moves aside from Rage
3. Toss all items or deposit them into the PC
4. Enter Lorelei's room in the Elite Four

You cannot leave Lorelei's room the way you came in, so you have to either defeat her in battle or black out. However, there is a problem with blacking out.

The only move that your Fighting-type Pokémon can use is Rage, which has a 100% accuracy and deals damage, then automatically is chosen on each subsequent turn. The only way to stop using Rage is to switch out Pokémon (which we can't do), have the Pokémon faint, or win the battle.

The AI in Generation 1 will always use moves that are super effective against your Pokémon's type. Lorelei's first Pokémon is a Dewgong that knows one Psychic-type move: Rest, which recovers HP. The HP recovered is greater than the amount of damage Rage can do. Thus Dewgong will never try to hurt your Pokémon and you will be stuck trying to attack it.

But what about Struggle? Well, it turns out your Pokémon will never do it. The first time you use Rage, 1 PP is deducted. But on subsequent turns, no PP is deducted when you automatically continue the attack. Thus you will never run out of PP and never use Struggle. Additionally, AI Pokémon in the first generation do not use PP, thus Lorelei's Dewgong will never run out of PP or resort to using Struggle.

There is technically a way out of this, but the odds are absolutely astronomical. There is a bug in Generation 1 where a 100% accuracy attack actually only hits 99.6% of the time. Therefore, there is 0.04% chance that your Rage attack will miss and you will have to spend another PP to attack with Rage again. However, this is only calculated on the first turn that Rage is used, not the subsequent turns. So if you ever successfully hit with Rage, no further PP will be deducted.

Because the base PP for Rage is 20 we need to miss the attack 20 times in a row in order to use Struggle (or if they have a PP up, more). According to the YouTube video "How to Escape Lorelei's Game Ending Softlock", the odds of missing Rage 20 times in a row are so infinitesimal that you have better odds of encountering 5 Shiny Pokémon in a row in Generations 2-5.

At any rate, if your Pokémon ever successfully lands a Rage attack against Lorelei's Dewgong, you will be stuck in an infinite loop and unable to end the battle.

Contrary to the accepted answer, it actually is possible to get stuck in Pokémon Emerald. See for example this YouTube video:

Game Freak do seem to have tried to think of everything, but they missed something.

It seems that the only way to get to Sootopolis City is to use Fly and you can't release a Pokémon that knows Fly there, or overwrite the move. However, you can also get to Sootopolis using Teleport. So go there using Fly, enter and exit the Pokémon Center, then Fly to Lilycove City, delete Fly, then Teleport back to Sootopolis.

There is another thing though. You must have a Pokémon in your party that can learn Fly, as it must know Fly to be able to get out of Sootopolis in the first place to reach the move deleter. And this Pokémon must still be in your party when you return, as to remove it would require visiting a PC, which would require visiting a Pokémon Center, which would set your Teleport location to that Pokémon Center rather than Sootopolis. So it seems they thought they had this covered. But, what you can do is use a specific Pokémon which can learn 5 different HM moves including Fly. Then after Fly is deleted, teach it a different HM move in that slot. Then you won't be able to teach it Fly, because doing so would overwrite an HM move.

You must also have less than 100 Pokémon in the pokedex, otherwise you could use Pokémon Box to import a Pokémon which can learn Fly.

You must also have only 2 Pokémon in your party and no Pokémon in boxes, as otherwise the game would allow you to trade with others.

TL;DR - it is technically possible to get stuck in Pokémon Emerald but only if you really are trying hard to do that.