For you non native english speakers, what snowgirl is saying is that using 'can I' technically infers 'am I able to' / 'do I know how to' instead of the intended 'will you ALLOW me to'. It is a very common usage, but not considered proper.
Edit: robert0n is absolutely right. "Can I" for "May I" is totally 'safe'. I think what I was trying to say is that if you were trying to be absolutely proper / formal / polite, you would choose "May I" over "Can I".
In Italian the use of posso definitely has a context of 'allow'.
Just to agree with the others here. "Can I?" in an English sentence such as this really does mean "am I allowed to?" even though it literally means "am I able to?". What I've seen in Italian agrees with gmcolletti in that "Posso" is just like the English: literally it means "am I able to" but in practice means "am I allowed to?".
@gmcolletti: I don't know about the qualifier you added about being "not considered proper". I will happily admit to having pedantic tendencies but I think "can I use (the telephone)?" has been pretty safe for a long time :-)
For what it's worth, Google translates "may I use the phone" as "posso usare...", but Reverso turns up both that and "potrei usare" as well as "permetti che faccia una telefonata?" I'm guessing "posso usare" is generally fine, but it would be nice to hear from a native speaker.