After establishing that over pressure in your tire will probably not cause a blow out one must address a big cause for flats – hitting a rough pothole (or a curb for that matter).
It is pretty clear how an impact of that sort can damage your tire and even your rim but the thing is that even if the car drives on without any evidence for damage some may exist and you will pay a price for it in a later time. If by damage to your car body, shock absorber, rim and so on.
You can see in the attached links some good explanations on what can be damaged, how to fix it and even a reference on having someone else pay for it (if a pothole was the culprit).
Now, I am not saying that after every bump in the road you should have your car checked by a professional but looking for damages yourself after a hit, being aware to what can be damaged and having your wheel aligned on a regular basis should be a normal thing for who ever wants to be safe and save some car money.