I think everybody has pretty much answered the question, but I'll add my thoughts anyways.

I don't know that I own any button down collar shirts at the moment, as I tend to prefer point collar shirts with a tie and that's usually how I wear any dress shirt that I'm going to wear. As others have said, it's certainly okay to wear a tie with a button down collared shirt...just know that button down collars are considered less formal, so it would not be appropriate in all situations.

Eagle JCS nailed it....the answer is yes and it depends on preference and the occassion.

As far as formal attire goes, the tux hire business and bad 80s fashion have totally muddied the waters on shirt-tie combinations. Because I'm an Army officer, I became aware of the difference between a turn down collar shirt and a wing collar shirt by reading the regulations...and then found out much more about it on this website:
http://www.blacktieguide.com/
I've found this website to be a GREAT resource for anybody wishing to dress in Black or white tie and do so appropriately and with style!! I know that it is common here in the US to wear wing collared shirts with black tie, but traditional style (as well as my uniform regulations) dictate a turn down collar, pleated front shirt for black tie and reserves the wing collar, hard bosom (pique cotton, heavily starched) shirt for white tie only. Just wanted to share that...a good piece of general knowledge that is commonly violated here in the US.