I've never had anything like this happen to me and I'm looking for advice as how to proceed. I traveled with another couple to Brussels. For the return home, the plan was for them to take the train to Amsterdam to fly back to the US from there while I was flying directly from Brussels. On the way to the airport, my friend called and said that our passports had been switched. We had never confirmed that the desk gave us back the correct ones, assuming they would (a 5-star hotel).
I ended up having the cab turn around to take me to the train station. I took the earliest (and fastest) train there in hopes that my friends would make their flight. (I'd already missed mine by this time.) They were able to reschedule with their airline (at no cost) and ended up taking a later flight the next day.
I wasn't able to change my flight and ended up paying for new tickets for me and my travel partner. (Total cost for train tix + flight was over $1000.)
The hotel said they weren't liable since we had several days to figure out their mistake and should have checked our passports. While that's partially true, the hotel lucked out in that we were able to fix the passport situation on our own since we knew each other. (Note that we booked separately at the hotel.) Had I received a stranger's passport, I would have had to go to the embassy and who knows how long I'd have been delayed.
I think the hotel should offer to reimburse me for their mistake. I'm at a loss as to how to proceed though. Thanks in advance for your advice!

I'm reluctantly leaning toward the hotel's stance, and this traveler's primary rule, check & check again to make sure everything's in order. You can, of course, continue to try to get reimbursed but for me, staying mad for who knows how long isn't worth the money you may or may not ever see. You might send your complaint up the chain or directly to the top and hope someone will be sympathetic, maybe at least for a credit. I agree that some compensation would be good PR on their part but ultimately we're responsible for ourselves, even mistakes made by others that effect us & can be corrected if we're paying attention, especially far from home.

That is a horrible situation. However, I think I am with MmePerdu, and the hotel. If you had looked at the passport when it was handed to you the issue would not have arisen, and there would have been no need to involve embassies if it had been a stranger's passport. Of course, if they had handed yours to a stranger who had left with it and were unable to give you the right one, you might have a better case. Did they give the passports back the day you checked in, or the following morning?

I agree with the above posters.
I'm sorry this happened to you, but I would have checked my passport as soon as they handed it to me. The hotel made a mistake, but as they said, it could have been handled immediately if you had checked your passport.

"Trust but verify" -- R Reagan