The bath is probably a mud job for that age house. Everything is set in concrete(mud). The floor will be set on between 4-6" of concrete. They installed cleats on the floor joists half way down. They then laid planks and tarpaper on top. They filled the joist bays with concrete and came up over them about an inch or so. They set the tiles on that. There are a number of ways you can redo the floor base. It depends on your skill level.
If the tub is dropped down it could be that the supports under it are rotted.
This sounds like you should gut the room and redo the bath. The amount of trauma and vibration you will produce to get the floor out will have the wall tiles falling off the walls.
This description is based on that this is the original bath and none of the walls have been replaced.
Ron

sounds like you have a bathroom gut job ahead of you if the tub sunk. I hope you have a 2nd bathroom. Tearing out the tile will involve a lot of hammering,using a masonary chisel and a hand sledge to bust up the tile. I would get that out first to see if they just tiles directly over the old tile or put a layer of cement board down first. The original tile is probably set in a mortar bed, I don't think they made mastic back then. There is a good possibility that there is no subfloor wood underneath the original tile. I owned a house built in the 1940's that has a tile bathroom floor. What they did there was put heavy tar paper between the floor joists and laid a thick mortar bed on top,leveled it and tiled over it. It was over an inch thick. I can't tell you how they did it but I can guess that you will have a thick mortar base under the original tile. You might also have floor joist problems if the tub has actually sank and is out of level. Sounds like you are going to have to gut everything out of the bathroom to rip up the floor and possibly fix the floor joists. Without seeing your bathroom, I can't say what you are going to encounter, but I would expect the worst and hope for the best.