I do it all the time. And I start sentences with 'and' too.
If you're not sure about using it, try a few alternatives then read them out aloud. If 'but' sounds more natural than using six other words to achieve the same result, then use it.
I am a great beleiver in reading my work aloud to myself it really does help spotthe rough patches.
In speech perfectly acceptable, In poetry equally acceptable in general narrative I think I would try and find an alternative. But some times you cant.
I'd say that unless you were doing a degree in the english language, then it's your work and write what you like. I use all sort of things I shouldn't, especially in dialogue if it fits in with the character. After all, how many of us really do speak the queen's english? Thankfully not many of us!!
Sue
The word 'and' is a very specific conjunctive and is defined as a 'joining word'. It's use to start a sentence should be avoided. 'But' is also a conjunctive and while the use of 'but' to start a sentence may not be an example of 'correct or good English' (as Ani and Sue have pointed out, this will fail you in any testing of grammar) it is one of those words where its 'correct' use is being challenged.
I think it may emerge as an acceptable first word but never as an example of 'good English'.
A former English teacher's answer: Yes. I don't know where/when the fussy supposed-rule slipped into common consciousness (it certainly wasn't around in the 18th century). But it's perfectly find to begin a sentence with 'but'.