I took my 2011 Ram 1500 and had oil change, 2 recalls fixed and had 4 new tires put on. Upon completion on everything but the 4 wheel alignment they came out and told me I had a TPMS sensor that was bad. Now on the way to the shop my sensor was working. I had a nail in the back tire and was checking it regularly on the way to the shop to make sure it wasn't loosing air/ going flat. All tires were reading fine. It's a direct system. I have had no false reading or anything that would indicate to me that something was wrong with it. I bought it in 2012 & it has about 164,000 miles on it. I just find it strange that I had no indication that something was wrong with it before they changed my tires and now I have a bad sensor. Could they have broken it? No they showed me nothing that says it had low battery or error codes, just came out and told me it was bad.
If you doubt their word, ask that they tun the test with you there. The tester is a hand held diagnostic tool; the mechanic walks around the vehicle and tests each wheel. It should be simple enough for you to ask to see and be shown the display.
I do realize this should likely go in comments, but I am not up to enough rep to do that yet. Soon.....
Since the answer is here, I will add too that just because the test shows that the TPMS is faulty, it does not mean that they didn't cause the damage.
Its hard to know the truth in situations like this when going by someones word but they do like to crack around the stems a lot. Especially if you have been putting air in it often from slow leaks. Visually you will not always see it and this usually will not throw codes but it is still broken and could blow out at anytime so the dealer definitely would not allow it to be reinstalled.
Those older sensors are awful. They can break just by looking at them the wrong way. Battery life is 7-10 years, so you're in that range right now. Tire dealers charge a lot less than car dealers for TPS sensor replacement. Recommend taking it to a tire shop for second opinion and replacement if needed.