Yes—but usually only to fill a missing party role
First, technically, any character who is played by the DM is an NPC, not a PC (DMG p.89):
A nonplayer character is any character controlled by the Dungeon Master. NPCs can be enemies or allies, regular folk or named monsters.
This idea of an NPC party member is covered on DMG p.92:
NPCs might join the adventuring party because they want a share of the loot and are willing to accept an equal share of the risk, or they might follow the adventurers because of a bond of loyalty, gratitude, or love. Such NPCs are controlled by you, or you can transfer control to the players.
DMG p.93 describes these as Adventurer NPCs:
If you don't have enough players to form a full party, you can use NPCs to fill out the ranks.
However, it advises that in this case, the DM may wish to to give control of these NPCs to the players.
DM-controlled party members are not without their problems. When the DM just wants to be a player as well as DM, what he's playing is termed a DMPC, and this is typically due to inherent conflict of interest.
It usually occurs with new DMs, who are nervous about playing D&D without controlling a character. Such a DM should remember that they control many characters, and do not need to play an adventurer.
In my experience, players reject NPCs who join the party, even exceptionally helpful ones. There's a sense in D&D that the players and the DM are opponents, and it seems wrong somehow for the DM to play both sides in combat. They may not entirely trust the DM's character, and plot betrayal is always a possibility.
In 5th edition, NPCs take a share of XP. This is likely to discourage players. Additional PCs will also make combat take longer, which is usually not a plus. Players take longer waiting for their turn, which isn't good. Even if you're careful to have the NPC be a support character (which is probably why he's picking a bard) and avoid stealing the spotlight, not all players are happy with this situation. Personally, I suggest that DM should only use NPC party members temporarily, if at all.