Become a MacRumors Supporter for $50/year with no ads, ability to filter front page stories, and private forums.
Yes, but only iOS 6.1.3 on the old bootrom 3Gs. A iOS 7 jailbreak maybe back ported to iOS 6.1.3 on the 3Gs and iPod Touch 4, but likely only those devices.
 
Please check iClarified for jailbreak updates

:apple:Can you jailbreak ios 6.1.3 or 6.1.4 unteathered and will it ever be possible?

Many people here are miffed that you haven't use the search feature to look for this answer, but to directly answer you:

You can NOT jailbreak 6.1.3 nor 6.1.4 untethered on any device right now.

Will it ever be possible? Well looks like there will be one when iOS 7 is released. The hacking groups have hinted that they are saving exploits up to crack iOS7.

You should bookmark this site: http://www.iclarified.com/jailbreak/ as it will immediately tell you when a jailbreak is out for your device. If you want to punish yourself, then subscribe to evador's twitter, but I find that maddening to try to decipher the stream of consciousness of those folks.
 
Register on MacRumors! This sidebar will go away, and you'll see fewer ads.