What crimes can a Thane commit without punishment?

I'm now a Thane in Whiterun - I was caught picking a simple lock in an inconsequential area and I was able to pull rank and tell the guards that they should remember who they're talking to, and they told me that they'd let it go but that I should be more careful.

Is that just talk, or can I get away with things repeatedly?
I'm guessing that I can't murder people and then claim diplomatic immunity, but can I always get away with low level crimes like picking locks or pickpocketing?
Is there a (daily?) limit after which I'm arrested?
Does it matter which guard sees me?

Aspiring supervillains want to know.

Solution 1:

I know this is an old question, but since I have not seen a definition of what happens here it goes. Basically being a thane means that the guards will clear a bounty of up to 3000 gold once per hold. It does not matter which guard sees you or how many crimes you have committed, so long as your bounty is less than 3000 you can have it cleared by declaring thaneship when the guard attempts to arrest you. Once you have used this privilege you will need to start paying the bounty just like everybody else.

See UESP article on Bounty - Declaring Thaneship.

Solution 2:

The crimes are:

• Murder
• Theft
• Murder of animals (e.g. chickens)
• Stealing (animals or objects)
• Pickpocketing.
• Assault (depends - sometimes you can get away with fist fighting!)
• Being a vampire (at the later stages - you get attacked on sight!)
• Transforming into a werewolf in front of people (attacked on sight)

And also, I'm pretty sure if you join the Stormcloaks, the guards of the empire will try and kill you.