To further elaborate on surviving Cliff Lee's loss with six relievers instead of being forced into a bench-shortening 12 man pitching staff, I'll toss out a concrete plan that could work with an eleven man staff.
The key is to be more organized with your bullpen. Let's assume a bullpen of the closer, David Aardsma, two short men in Brandon League and Mark Lowe, and three longer relievers in Shawn Kelley, Sean White and Rule 5 pickup Kanekoa Texeira.
We commonly accept using five starting pitchers in a set rotation. This goes back decades (though farther back they used 3-4 pitchers instead of five). This helps define a timetable for each pitcher, as they know what day they have to throw on and that they will get days off to rest... and sets a mostly definitive workload: They know they will throw up to 100-120 pitches every five days.
Why not use your relievers in a rotation of their own?
The only man that will not work on a schedule is Aardsma. In the traditional closer's role, he will come on to lock down any game with a 1-3 run lead, preferably to start the 9th but sooner if needed. On occasion, he will step into a tie game in extra innings.
Otherwise, take the two setup men and work them on alternate days. Lowe is on standby to work the first day. The next day, Lowe gets the night off and League is on standby to work. Flip flop, rinse and repeat. Now, on a given night it's possible the designated short guy may not be needed. Felix could go eight strong innings and Aardsma can just take the ball in the 9th, or Felix could finish the game himself. In that case, it's your call whether to use the unused setup guy the next night or skip him turn and call on the other guy right on schedule. This setup guy will typically work the 8th, or 9th in a non-save situation. He'll be expected to face 3-6 batters, maybe less in the right situation.
The other three relievers? They will work in a three man rotation of their own. Each man, on his designated day, will take the ball when the starter is done, whether in the 7th or the 5th, and face up to the entire lineup a single time (up to 9 batters faced). Some days, he might only have to face 2-3 batters. Some days, he may need to face the entire lineup once. Rarely will it be bad enough that he faces nine guys and isn't at least part way through the 7th, if not to the 8th.
Obviously, this will usually require a lot of work for a reliever in this rotation, so each guy gets at least two days off following his outing. For example, Kelley will work one day, then Texeira the next, then White the next, and then back to Kelley. Rinse, repeat.
This schedule can be adjusted if need be. For example, maybe a middle reliever isn't needed on a given day, as Ryan Rowland Smith threads the needle for 7 innings so Wak goes with Lowe on his scheduled day in the 8th before turning to Aardsma. Thus the unused reliever can go the next day and give the other middle relievers an extra day off. Or maybe Ian Snell gets shelled the next day, and both the previously unused reliever and the middle man that was originally slated to go on Snell's day can each face 9 batters to bridge the gap. This would probably shorten the rest for one guy if you stay on the middle relief rotation, but it can typically be managed. A great start from another pitcher or a timely off day or rainout can get you back on track, or to a lesser extent a starter goes 6.1 innings and the overworked middle guy only has to work a couple batters or something.
A lot of dominos would have to fall for this plan to fail enough that you have to cave in and call in a 12th pitcher. If a guy gets hurt or something, DL or option him and call someone else up. Tacoma should have a fair share of options (Luke French, Garrett Olson and a couple others) than can step into either the middle relief rotation or setup rotation.