For me, I see obvious conflicts and misunderstandings occurring between E vs. I types, S vs. N types, and T vs. F types, even if they have two or three other preferences in common.
I certainly see it occurring in J vs. P types as well, and I've read all about these in books about type. P's feeling stifled by J's, J's viewing P's as unorganized, etc.
But after thinking about it some more, I think I've noticed that I clash with J types just as much as, and actually probably more than P types. I guess it all depends on the situation and person of course...there are some situations where P types drive me up the wall.
But I really see J types clashing with other J types just because both sides seem like they're more likely to be stubborn and less likely to budge on whatever it is they're feeling at the time. Of course, if these two things align, that's great...but if they don't, it feels like both types will be less flexible.
Of course this is a hard question to answer, since it will greatly depend upon the exact type of the person, other attributes of the person, and the situation at hand. But if we're just speaking as generally as possible:
J types can either answer personally or talk about J's in general, I guess P's have to talk about J's in general.
However, you'll notice I haven't mentioned P's clashing with P's. This certainly can happen, but I'm assuming out of the three pairings, it happens the least. However, if any P type wants to chime in giving a different perspective, I would find that interesting as well.