I purchased an 06 Volkswagen Passat four months ago and the engine mount just broke, throwing the timing belt out of whack and causing the valves to hit the pistons. It's no longer under warranty. Should/did the dealer see this coming or is this just really bad luck?
Dealers aren't psychic. They sold you a car that ran fine for four months. Nothing was broken so you can't expect them to fix it?
Every manufacturer has "interference engines" which are the ones that develop that valve damage. Most mechanics would never own one of those cars, but the average person has no easy way of knowing which ones those are. Some cars, Hondas for example, recommend the timing belt be replaced every 75,000 miles, and they were known to break at around 65,000 miles. When you buy one of those cars used, both you and the dealer have no way of knowing if the timing belt was recently replaced by the previous owner unless that person mentioned it.
To give you a better answer, yes, it's just bad luck the belt broke, but there's more to the story. Worn engine mounts are fairly common and will not cause a timing belt to break. If something more catastrophic happened to the mount, that would really be out of the ordinary, but it is still unlikely to cause a belt problem. Those belts are just too well protected.
Here's a different thought, and I don't mean this sarcastically even though it might sound like it. What we used to tell customers when their tv broke down was it might be worth $100.00 on trade toward a new one when it was still broken, but if they spent the $75.00 to have it repaired, it would only be worth $125.00 for trade-in value. Cars are the same way. If you trade your car in with the broken belt, you will get almost as much for it as you would after you pay for the expensive repairs. This might be an opportunity to trade the car off unless you really like it. Volkswagen and General Motors have a real lot of customer-unfriendly business practices. You will have a huge repair bill including towing to the dealership if you simply let the battery run dead or disconnect it to replace it. The only reason to design cars like that is to squeeze more money out of their customers after the sale. Sadly, whatever GM dreams up to cost people money, most other manufacturers copy a few years later. I've never worked on or owned a Hyundai, but at least right now they are the best company when it comes to being customer-friendly after the sale.
The valve job is going to cost more on a Volkswagen than on most other car brands but it's not even close to the cost of a total engine rebuild. I'm frustrated to say that repair is all too common today. Head gaskets, among all brands of cars, are another real common problem that we never saw 30 years ago. Basically, you will be getting a new head gasket, a common repair, along with new valves to replace the bent ones. Of course you will have a new timing belt too so you should have no worries for another 50,000 miles at least.
As for the dealer ripping you off, put that idea out of your head. There is no way they could have any idea what is going to happen four months later. Even if they had guessed at what might happen, they could have replaced the timing belt, fuel pump, two tires, and the power steering pump, and they still might have guessed wrong. Typically a conscientious dealer will perform a safety inspection, repair what is known to be needed, perform an oil change, and wash the engine and interior. Those are things they expect to do to every used car they sell. Any expensive repairs beyond normal maintenance either raises the selling price of that car which makes it less attractive to potential buyers, or they are going to offer you less trade-in dollars for your old car.
You didn't say how many miles are on the car, so you might consider asking the Volkswagen dealer if there is any warranty left through the manufacturer. If not, you might find the repairs to be less expensive at the dealer than at an independent repair shop. Volkswagens require a lot of specialty tools that most shops don't have, so they will typically remove the cylinder head and take it to an engine machine shop. That adds to their cost of repairs which gets passed on to you. The dealer's service department will have those tools already, and with their experience, the repairs will probably get done faster. They also have an interest in doing anything possible to ease the pain and make you happy if they think you might be a future customer.
One more thought; from working at a very nice Chrysler dealership years ago, I learned they have a set number of dollars they can spend every year to perform repairs under warranty on cars that are just out of warranty. It's for customer satisfaction even though they aren't obligated to do that, and it's up to the dealer's discretion where to spend those dollars. If you bought your car from a Volkswagen dealer, they might offer to help with part of the repair bill, especially if it has low miles. Typically they try to help out their regular customers and people in need. Crabby or angry customers don't usually get any help. I have to be careful to include one more comment about this special fund. The service adviser might make some comment about "we got this part handled under warranty for you", but you should not take that to mean that part of the repair would be covered under warranty for everyone else's car. They won't tell you about that special fund; they'll just use it for your benefit but other people will have different circumstances and may not get the same consideration. If your repair shop is able to help out in any way, a bag of chocolate chip cookies a few days after the repairs are completed is always appreciated.
Sorry that you had the bad luck, but it's all too common with newer cars. I read about these high-cost repair bills over and over on this site every day. In part, these designs are responsible for the higher power, lower emissions, and better fuel mileage from such a little engine, but like everything else, it comes at a price.