Some tp's like people need to get a little age on them before they blossum.

Check your fertilizer to see if your first number is higher or equal to the second. If it is change fertilizer. I am making an assumption that you use solid rather than liquid in the fall. If you are using liquid change to a solid.

You may want to do a light feeding just after your normal blooming season.

You may be able to see next years buds start to swell sometime this month or next. Some types will not swell until late winter.

We also have a beautiful (foliage) peony that doesn't bloom. She bloomed a couple of years, then stopped. My only clue is that, since she's planted right below a bird-feeder, and we've managed to attract a WIDE range of birds to the yard, they might be eating the ants? Would that make a difference?

ptilda, the ants have nothing to do with blooming or not blooming.

Some possible reasons for established peonies failing to bloom are -

Buds were killed by late frosts.

The soil is too dry during spring growth cycle.

Excessively hot weather during bud development.

Too much nitrogen fertilizer.

Under the bird feeder they are definately getting too much nitrogen. Move the bird feeder and some of the soil surrounding your plant. Keep the soil because after it composts it is very good for use but right now it has too much nitrogen in it.