The recent DTV switchover had me musing that theoretically a TV made in 1941 (when the 525 line standard began) could still be in working order until 2009. Of course such things as tube burnout and the desire for higher quality pictures probably means a TV really didn’t exist outside of museums. What is the oldest you have or know someone having?
My uncle still has a TV that has Channel 1 on it. I don’t know what year it was built, but it was the first TV in our hometown, and (to the best of my knowlege) is still in mint condition and still works (although with the digital revolution, there’s nothing for it to pick up anymore).
Ha ha. I have a 1947 Stromberg-Carlson TV-12. It also has Channel One. The television, radio, and phonograph are all in working condition. Of course, I also have a wonderful boyfriend who restores antique televisions for a living. 
I have two other sets also in working condition, although they are from the fifties.
For some interesting reading you can check out the website of The Early Television Museum. We went to the annual convention in May and there was a lecture on a set from 1931 that had been fully restored. The picture tube gave off a green cast. It was absolutely amazing to watch it in action.
Note: You can hook up a converter box to any early television just like a new one. Most of the time when displaying our sets we tend to hook them up to DVD players. I, for one, like to watch Harry Potter movies on my antique sets. 
We have a 1956 Philco Diamond. Great sound, needs a picture tube. Would love to get it fixed without too much money – thanks for the link, Pie!
Moses Znaimer has a massive collection of early TVs as well, and his new museum is opening to the public in July - virtual tour available at this link.