So, I just talked to Bunnie over and over and over again. Eventually she said something along the lines of "Umm... I've used up all my brain power, come chat again when I have something clever to say." And when the conversation ended she walked around with the thinking bubble & dots above her head, with her head lowered.

I thought peppies couldn't get tired of conversations, but is this not the case?

Nope, all villager types can get tired of conversations when spoken to enough.