Wendigo
It also occurs to me that the tampon, while it's unlikely to stand up to two pounds per square inch of pressure from one direction, being a wad of cotton, is on the other hand capable of transferring foreign contaminants into the wound. It's also basically an enormous blood clot, which poses its own risks:
http://en.wikipedia.org/wiki/Thrombosis
Soakin' up a nosebleed, as suggested above, or in its intended function, it's not actually in the bloodstream, just in a place where blood has collected. Its new position in the bloodstream is not super appealing.
I think I'd stick with the tourniquet, where possible, and just forget about the tampon.
They're made from cellucotton, the same thing older field dressings were made out of. There was nothing wrong with older dressings, newer ones are just high speed.
Our medics say tampons work, no one has ever denied tampons work, and we've only heard good things about sticking tampons in a wound. No one has ever died from a tampon being stuck into them, and it has saved lives before.
Really, it's the exact same thing as packing a wound with gauze.
WARNING: Possibly disturbing medical image
Except that you don't need to stick your fingers in the person. You just jam the tampon in. When they get to the medic or CSH, they take tampon out and the medical professionals replace it with actual, medicated gauze.
As I said before, if the bleeding is really bad, go to is tourniquet..
But lets say someone is hit in the shoulder by a 5.56 round and I'm the CLS. First thing I'd do is make sure there's no danger to me, then I'd run out to him. This part should be done in less than two minutes and only if I have a really, really solid base of fire covering me. I check for massive hemorrhaging first, because that's what's most likely to kill this guy. I see the gunshot would and immediately stick a tampon in it and check for an exit wound. If there's an exit wound that's bleeding, I'll have to pressure wrap it. Can't stick a tampon into an exit wound, don't have time to pack gauze, tourniquet won't do any good there. Finish checking for bleeding, then I check for breathing, head injury, and circulation pretty much all at the same time. That should all take me about two minutes. Pick him up, take him back out of the combat zone. Once he's to safety, check circulation, breathing, bleeding, shock, fractures, burns, head injury. If it's obvious that he wouldn't have fractures, burns, or head injury, I can skip these three. At this point, breathing and circulation are a priority over bleeding so I check that first. I recheck my pressure wrap, determining everything else is fine and preparing the casualty for shock. Let's say he's still bleeding. Oh s**t. I'd take my thumb ad press it down into his shoulder as hard as I could, hoping to get a pressure point and stop the blood from the artery at least. I may not be able to stop all the bleeding, but if I can hold off that main artery in the shoulder long enough for the medic to get over and work his little EMT magic, I may just save this guy's life.
Moral of the story, the tampon isn't going to fix a gsw. It may end up being pretty much useless.. but we may not have the time or ability to do anything else. In this case, I needed a pressure wrap because exit wounds will generally be too big for a tampon. Well, that's okay. The tampon is still helping the bleeding in the entry wound. That's all I need to keep him from dying in the next three minutes that it takes me to get him out of danger. Now that me and the casualty are no longer being shot at, I recheck the wound and see that the tampon and the pressure wrap aren't stopping the bleeding. I'm no longer in threat of being shot, so now I can safely apply pressure and wait for the medic. The tampon is a quick fix before some higher form of aid is available.
Next time we do a CLS class, I'll be sure to ask the medic again just in case.