Can the bard play music or musical instruments?

Sadly, no playing. There doesn't seem to be a position like arch-mage to aspire to. Caveat: I haven't completed every single quest from every bard that has one, although they seem to fall into the "Help citizens of XX" group of miscellaneous quests.

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.

There's no music playing for the Dragonborn.... unless you want to.

As the other answers have said, the Dragonborn cannot sing or play in the base game. However, if you're in to such things, there is a mod called Become a Bard by Pevey.

Tired of being a warrior or mage? Become a Bard!

• This mod lets you play all of the instrumental and voiced songs included with the game or choose from over 125 new songs added by this mod. You can also add you own music and play it in-game.

• You can pick up and play any instrument in the game, including the war horns.

• To play a song, equip a musical ability in your spell menu. Press the shout key to use the ability. Press the shout key again to stop playing.

• Each time you perform, your Bard Skill will increase. See your stats in the bard journal in your inventory.

• Your follower will play along with you to any song with two parts. Some NPCs around you might applaud. Applause becomes more enthusiastic as your skill increases. NEW: there is more information about each song in the songbooks themselves, including the song titles, number of parts, instruments required, and song length.

• You can go up to any innkeeper and tell them you would like to perform. The tips will be minimal (10-25 gold without perks), but it should be enough for a meal, and you also get a free room for the night.

• Once you earn one of the bard perks (see below), you can request to perform for a jarl. Jarls do not pay money, but if they like your performance there is a chance they will give you a gift (random enchanted item).

• If you complete the Bards College quest line, your Bard Skill is high enough (50 or higher), and you have earned one of the bard perks described below, Viarmo may have additional work for you.

• Each time you play, your Speechcraft skill will increase a tiny bit.

• The volume of your songs is adjustable in the normal Skyrim options menu under Audio. Adjust the very last slider, Player Songs.

Personally, this mod is so good that it's part of my "permanent load list". You can find this mod on Nexus by Pevey for Oldrim and SSE. And you can find it on BethesdaNet by Pevey for PC and XBox One