Could be Botrytis. I had one that refused to bloom for two years. The buds would form and then turn black, wilt and sometimes the entire stem would just die back to the next fork. Someone told me it might be a fungus called Botrytis.
It seems to be one the more common problems with Peonies.

When I researched it, the easiest way I read to get rid of it was to cut back the plant to ground level in fall, before it dies back, so I did. I also removed all the mulch from around the peony(about a 5 foot circle) and threw it away. This year I have about 6 monster buds on that Peony ready to open next week I would imagine.

I think this link might have a few other options for control.

Here is a link that might be useful: Botrytis on Peonies

i don't know what peonies like, but are they hungry? how's the drainage? the one in the middle might be getting all the goodies. what's on the outside ends of the row? I have a rose bush sulking for that very reason - can you believe i made the drainage too good and now it wants more water!!!

Last year I put epsom salts around mine (forget who told me to and why) and this year they are chalk full of buds, not sure if that is what did it or they just needed to get established this is the third summer and the first one that I should have blloms, last year I had 3 buds but they never opened.