Tournaments have a lot of variance, so playing them will require a large bankroll. Most people throw out 4% (25 buy-ins) as a standard bankroll number, but serious players are going to need more.
A 4% tournament bankroll is fine for casual players. If you have $300 online that you can easily afford to lose, I wouldn’t expect you to play less then $5 tournaments. If you lose a few buy-ins though, you should be safe and drop down to $3 or $2 tournaments. That way, you’ll never burn through your bankroll.
Mid to high limit and serious tournament players should be much safer with their bankroll. There’s nothing worse then going from a $200 tournament bankroll to a bankroll for $20 tournaments because of negative variance.
The optimal tournament bankroll is 50 buy-ins. This way, losing a few tournaments won’t hurt you, but scoring a nice win can still have a large impact on your bankroll size.