Can the bard play music or musical instruments?

Solution 1:

There isn't any purpose. There is no way to play music in game. You don't get a room in the college or access to anything special.

As far as I can tell, the only difference after completing the Bard's College quests is that some guards will refer to you as "Bard" occasionally.

Solution 2:

Actually, the accepted answer is technically wrong. I say technically wrong, because it's a bug that means that completing the bards quest does nothing, when actually you should get a bonus to your speech learning.

You're supposed to get the Active Effect 'Gift of the Gab', which lets you develop Speech 15% faster, but it doesn't activate because of yet another bug.

See the bugs section here. Hopefully this'll get fixed in a future patch and will work retroactively for those who've already completed the questline.