I wish I could say that this is a rare phenomenon, but sadly, it is quite commonplace. There are still several options that anyone can choose from, to settle their dispute outside of court and avoid litigation.
Don't Give Up/Give In- Just because the other party refuses to mediate, does not mean that they are unwilling to settle. This just means that you have to get creative. Do not assume that you have to throw in the towel and hire the local 'pitt bull' of a litigator to settle your case. Should you decide to do so, I would only recommend litigation as a LAST resort. There are several other forms of alternative dispute resolution you can try before relinquishing your power (and retirement savings) over to your 'pitt bull'. Here are just a few ideas:
a) Arbitration- This is a process by which parties select a mutually agreed upon third person, usually a retired judge or attorney, to help them settle their dispute. During arbitration, the arbitrator will listen to both parties and offer their 'objective' assessment of the dispute in question. They will then determine what an appropriate settlement would look like for both parties(you might liken this to a judge rendering a verdict, but outside of court). If the arbitration is legally binding, the settlement agreement can be upheld in a court of law, should the other party choose to contest the decision.
b) Conflict Coaching- Just because the other side refuses to settle, doesn't mean that you have to. You can always meet with a Conflict Coach, a trained Conflict Resolution professional, who will help guide you through any dispute. He/she can help you communicate with the party, de-escalate their behaviors, deal with impasse, and ultimately settle the dispute, all without ever having to meet the opposing party.
c) A Sit Down- There is this primitive thing that most parties forget to do when faced with a dispute, talk to each other! Many of the parties I come across in mediation, are not only avoiding communication with the other person, but have often hired attorneys and/or taken the dispute to court BEFORE ever attempting to speak with the other party! Never underestimate the power of a good sit down discussion, without mediators, attorneys and/or arbitrators present.