I have a good friend who is worried quite a bit about her labor stalling and then ending up with a c-section. So ... this post goes out to you!
It is quite common for labor to stall at some point. Anxiety, exhaustion, dehydration, frustration, poor fetal positioning, and lying still for long periods of time are a few contributing factors. Some women just have weaker contractions and are in some sort of labor for days. Some women have not had any sleep the last few days or weeks, and so their bodies will take breaks to try to reserve energy for pushing. Whatever the case, if the mother is coping well and the infant is tolerating the contractions just fine, there is no reason to immediately turn to a c-section.
The only problem is, some women could be in labor for days. This is far from appealing to anyone. The mother and her partner will be emotionally and physically exhausted, and the care provider will not only be tired, but they will be losing time they could be spending with other patients. Everyone has their limit that they are willing to let things go naturally, and you will have to decide what yours is. You should definitely be aware that hospitals have the shortest amount of time that labors are allowed to endure, especially as it seems so very much easier to wheel down the hall and cut the baby out. Most hospitals have a 24 hour policy; from the moment you check in you are on a time clock. You will also be likely to hear people saying things like, "Let's get this kid out on my shift!" or, "Your doctor will be here in a couple more hours," which is kind enough, but does definitely put pressure on the mother to achieve some sort of schedule.
Birth does not happen that way.
I would suggest making a birth plan, having a good idea of what is most important to you, and then once labor starts .... just don't worry about it. Let your partner handle schedule stuff while you go inward ... get in that mammalian brain that actually does know the best way to get that baby out. Sometimes the less you think, the better.
Here are a couple of good articles: