Here is another idea for purposes of learning the cause of the leak. Get some good stretchy packing tape such as Scotch. Bridge the gap between the tile and the tub. Take care to press into the grout lines. Possibly do a second layer higher than the first.
Take some showers. If the leaks stop, figure the leaking was at the tub-tile joint.
Maybe there is a better tape for the purpose. You might want to test how removable the adhesive is before a full scale test. I don't know if a masking tape (which remove more cleanly) would be water resistant enough.
ok, here are some data relevant to this question.
When you go to the caulk section of the hardware store (at least the one in my town), one of the things they have is a special kind of tape for this purpose. It is very thick and kind of squishy, almost like a flat hose. It is marketed as an easy way to caulk. Since I was having to recaulk every couple of months, I thought this was the answer to my dreams. I think mostly due to the nature of the tape stuff itself, it really wasn't a solution, at least not for any length of time. The tape couldn't really get in there and seal up like caulk does. With use of the shower, the tape didn't really stick well.
I forgot to mention in response to one of the questions above, at a different point in time, I also thought that backer rod would be the answer. I cleaned everything out, put backer rod in, and recaulked over the backer rod. In the end, believe it or not, the backer rod ended up making the situation WORSE. I don't understand why. My best guess is that people getting in and out of the tub again loosened the caulk. The backer rod stuffed in there may have increased the size of the crack between the tub and the tile wall ---> more water in basement.
After the backer rod episode, a brilliant hardware store guy came up with another idea. I bought a short shower rod, one that fits across the front of the tub underneath the hot and cold water handles. I cut a piece of waterproof shower curtain to fit across this rod and come down about 3 inches. In other words, a teeny weeny shower curtain over the front edge of the tub where it hits the tile wall. There are a lot of inconveniences with this arrangement. However, my impression is that when everything is right, it does stem the water from coming into the basement. One of the problems is that water can kind of get behind this thing from the side front corners of the tub. Another problem is that it is one of those shower rods that is held up by pressure (by a spring inside), so it slips and then is no longer tight against the wall. It is off at the moment, but I will put is back up to retest. My impression is that it can help the situation.
Thanks again for all your help considering all of my strange leak problems,
Ruth