There are basically two possibilities here in which a discharge petition happens, and neither is likely. Why? Because the House leadership really, really, doesn't want discharge petitions to become common. And the rank-and-file share an interest in it, too.
One is that Speaker Boehner and most Republicans really want a bill to pass, but they don't want the blame. That means the eventual floor vote would have the bill passing with mostly Democratic votes, so the only question is the path to get it there: either Boehner can bring it up despite his promises not to (presumably after some event "changes" things, such as a House-authored bill going down in flames), or they can give tacit approval to some Republicans to sign a discharge petition. He'd rather have the former. Sure, that means that he has to take the heat, but that's what you get for sitting in the big chair.
The other is that Boehner and most Republicans really oppose having something pass. In that case, the Republicans who signed a discharge petition would be seriously undermining what the rest of the Republicans want.
The key here is that House leaders strongly want to maintain party discipline on procedural votes, and a discharge petition is a very strong violation of party discipline -- even stronger, really, than a vote against a "rule" on the House floor.
The power of the majority party in the House basically comes down to agenda-setting, exercised in part by committee chairs but most critically by control of the House floor through the Rules Committee.
A "Hastert" violation -- putting something on the floor that most Members of the majority party oppose -- doesn't cede control of the floor. A discharge petition does.
Therefore, it's far more likely that if they want something to pass, they'll put it on the floor themselves. And if they don't, it's unlikely that rebels would sign a discharge petition. It's one thing to vote against the party on substance; it's a much bigger deal to work with Democrats to gain control of the House floor.
Granted, for supporters of the bill, using the possibility of a discharge petition to apply pressure to Republicans who support a bill makes sense anyway. But the practical effect would almost certainly be having those Republicans push Boehner to bring the bill up, not to have them actually sign on.
So the existence of the discharge petition procedure is a lever that bill advocates can use. But ultimately, that's probably not how a bill gets to the House floor.