It's been done, not terribly common but more so than you'd think.
I can't think of a team that has ever employed a 6 man rotation year round but there are a LOT of examples of teams who have done it for a stretch of 1-2 months including the Angels, Astros, Mariners, Nationals, and the Phillies.
It's not ideal, but it's probably easier to find bullpen arms ride the shuttle and ideally when you have an extra day of rest you can go an inning longer, partially negating the need for the bullpen.
I'm certainly not claiming that is the only way to go, but something will give if innings are not managed with our starters. I'm open to suggestions to sway my opinion.