This is a difficult question to answer, no matter what was said or done. The first thing to remember is the advice Jesus gave Peter about forgiving someone who has sinned against him: “Then Peter came to Him and asked, ‘Lord, how often should I forgive someone who sins against me? Seven times?’ ‘No, not seven times,’ Jesus replied, ‘but seventy times seven!’” Forgiving the other person in the relationship should be the first step you take in deciding what to do.
You can think of dating/courting as practice for marriage. This doesn’t mean you should stay with the first person you date/court and marry them. God may lead you to someone else, but it is best to forgive and try to work things out unless, through careful, humble prayer, God has led you to leave the relationship. You should put into practice the idea of making things work, as you would in marriage, instead of running away when your potential spouse does something that bothers or hurts you.
There is one thing that would definitely be a reason to consider “breaking up” with your boyfriend/girlfriend. Paul tells us in 1 Corinthians 5:9-11, “When I wrote to you before, I told you not to associate with people who indulge in sexual sin. But I wasn’t talking about unbelievers who indulge in sexual sin, or are greedy, or cheat people, or worship idols. You would have to leave this world to avoid people like that. I meant that you are not to associate with anyone who claims to be a believer yet indulges in sexual sin, or is greedy, or worships idols, or is abusive, or is a drunkard, or cheats people. Don’t even eat with such people.” If your boyfriend or girlfriend indulges in sexual sin continually—or any of these sins continually—Paul says we are not even to eat with such a person. Perhaps if it was a one-time offense, it may not be something to leave them over. You have to realize, the sin is not against you, but against God. God instructed Hosea to love his wife (who was a prostitute and committed adultery against him) as an illustration of how God still loved Israel after they had turned from Him (Hosea 3:1).
Circumstances are different for each situation, and it is always difficult to tell someone what to do in a given situation. It is clear that God wants us to demonstrate forgiveness in all situations, but as for leaving someone, the only way to know how to handle a problem in a relationship is through prayer, with discernment, wisdom, and compassion.