No, but not for the reason you might think.
It's a common play to announce the trainer you're playing, put the card in the discard, and then do it. This is actually, technically, the wrong way to play a trainer. It doesn't really matter all that often, but it's very important for Puzzle of Time and cards like it.
When you play Puzzle of Time (or any trainer, really), the correct way to do it is to put the trainer in question somewhere in play that is not your discard pile. Then, once you've fulfilled its conditions, it goes into your discard pile.
When you think of things like this, your question becomes very easy to answer: when you play Puzzle of Time, the card hasn't yet reached your discard pile, so you cannot simply pull an endless loop of them.
Now, with that said, if you had one in your discard pile and then played two more, you could take the one from the discard that is not part of the pair you just played.
