Tire Question

Would you run an odd brand tire on your rig? Recently, while camping, I did a visual inspection of my tires. The passenger side rear (two axles) was beginning to separate. This is my second set of Maxxis and the first problem I've had with them. I found a local dealer who was able to order one and it arrived the very next day. The first dealer I called said it could take three to four days to get the tire in. Either way it was not a major problem because we weren't scheduled to leave for a few days anyway. Here's my question: Let's say you're out on I-40 and you have a tire go bad. You're running Maxxis, or Goodyear's or whatever. The dealer can't get you a new tire for three or four days but he does have in stock another reputable brand. Can you run an odd brand tire if it is the same size, weight rating, ply, etc.? I had never thought about it untill this happened. (Just in case anyone is curious, my tires are four years old this month, and have approx. 4,000 towing miles on them. I'll be replacing the other three soon, but I wanted to wait till I got home and use my regular dealer)