I recently overcame one of the more frustrating & lingering problems in our 50s house and wanted to share what I found in the hopes that it might spare someone else countless hours and $$$. Our hot water is provided by a tankless coil in a oil burning furnace. From this coil, all of the hot water is fed through the house off a single 3/4" line. From the day we moved in we had great hot water everywhere except for the tub in one bathroom. You would turn on the hot water and it would heat up nicely for the 1st 60-90 seconds, but then it would go cold - much to the chagrin of kids & guests. This problem made no sense - the water got hot initially, and the fixtures right before (sink in the same bathroom) and the right after (shower upstairs) in the sequence of the feed line had no issues. We were stumped and so was the plumber we finally called in. After dealing with this for weeks we very randomly stumbled on the issue - we were turning on the hot water TOO much. Essentially, by cranking the hot water valve all the way open we were drawing such a flow rate that after the initial couple of gallons that had been sitting in the coil were exhausted, the water coming through behind it wasn't spending enough time in the coil to heat up. It was brilliantly simple once we found it, but terribly counter intuitive until we did - who turns down the hot water to heat it up? Since finding the source of the issue we've been playing with the behind the wall adjustments and the pressure regulator coming out of the furnace, but that's all icing on the cake once you have hot water.
I racked my brain for hours on this one, hopefully it saves someone in the future.