Is there a limit to the amount of xp you can have?

If you try to gain to much xp through commands the xp bar will dissapear, as well as your level on your HUD. This shold to occur around level 32,767.

The maximum XP that you can earn from the /xp command is 2,147,483,647 levels. The maximum experience level is 24,791 (With just /xp).

Without glichess or additions to Minecraft the maximum xp level is 32,767 this is using the /xp #L command, where /xp is the level and L increases the person's level.

For more info on experience visit: http://minecraft.gamepedia.com/Experience

Yes, there is a limit.

The level limit obtainable using the /xp command is 32,767, which is signed 16 bit integer.

The maximum level you can obtain without the XP command is 24,791.

Source: Minecraft Wiki's experience page

However, like most limits in Minecraft, it is near impossible to the limit. Bottles o' Enchanting gives 3-11 XP, and Chicken Jockeys gives 22 XP.

You'd need use 920548415 to 251058658 Bottle o' Enchantings, or you'd need to kill 125529329 Chicken Jockeys to get to level 24,791. (Or somehow find a way to respawn Ender Dragon, you'd need to kill 230137 of them)