This has apparently been an issue on earlier versions of LinuxMint with Cinnamon but using the regular launcher, although I can't confirm this.
The problem seems to point more towards Cinnamon, but I couldn't find anything relevant on the matter on the LinuxMint forums.
I've tried the Stock Launcher/FTB/Tekkit/MultiMC and all of them allow me to full screen on Cinnamon.
I've also tried an openbox/Gnome Classic/XFCE session and they also allow me to full screen. This bug happens regardless of the "desktop effects".
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@tails, I hit F11 and it tries to fullscreen for a second, then goes back to normal size, no crash or anything.
New development: As I was typing this message I tried to go full screen and it worked for the first time ever, it seems to be intermittent at this point.
This is a fairly new LinuxMint install, so I haven't had the chance to mess anything up.
Nothing out of the ordinary in the console output as I press F11, this is very bizarre.
Please see this video: http://youtu.be/c-YZoIebYiU
This is very likely a LWJGL issue, feel free to report it on: https://github.com/LWJGL/lwjgl/issues
There is very little we can do.

Yeah but Erik, this only happens with the Dev launcher, I use LWJGL 2.9.0 on the old launcher/FTB/Tekkit/MultiMC and it does not do that.
If you tell me that it's an Issue with my Desktop Environment that's fine, I was merely reporting it.
It's not that big of a deal, I just log out and use another DE.
This issue will persist from 1.6 onwards for you, you might as well report it (it's pointless for me to do so because I cannot actually provide them with a setup to debug it).
We create the window differently for 1.6 (and thus in all snapshots after 1.5.1), and this is very likely a LWJGL issue (or maybe even a java one not 'guessing' behaviour of your Desktop Environment right.

Moved to project Minecraft, since no launcher issue

Duplicate of MC-2067.

I'm not sure this is a duplicate of MC-2067. On my Fedora 19 system, 1.5.2 has no problems with fullscreen, but 1.6.1 gets me behavior described here.
Also note that the behavior is slightly different – in that bug, the pause screen comes up, but here, it doesn't.
Please explain what do you mean by cannot full screen. Does it crash? Then please attach the console output / crash report.