Yes. It happens all of the time. Some people simply do not believe in divorce no matter what happens. Others have gone through therapy as a couple have come out stronger for it. Some betrayed people have found a way to forgive if they believe in their heart that their neglect in the relationship may have contributed to their mate seeking other options.
Generally speaking what happens for couples that survive is the cheater (sincerely) regrets her or his actions, the hurt party remains "in love", and neither spouse wants to get divorce. They start working on building up trust between them and take one day at a time.
Having said that sometimes the betrayed party (instantly forgives) simply because the thought of being cheated on and going through a divorce is too much for them to handle.
Later on after stating they forgive they realize that they can't even if the cheater is bending over backwards to make amends. This is why it's important that the hurt person (take a break) to decide if they can in fact forgive. The only way to forgive someone is to NOT dwell on what they did wrong but instead focus on the things you love about them. If every time you see a person you think about them cheating on you then you will never move past it.