"A tenant reported that a light was out in their apartment." You need an electrician. You should not be attempting this work yourself. Just for mystery's sake: How many wires are in the switch box? 3 or 6? If 3, that white is not a neutral but leg of a switchloop. To use your test light to see if power is getting that far, you would connect 1 end to the black and the other to ground. If that lights up, then the problem is likely at the light fixture.

Also, it isn't going to be easy to tap, even if you have a cable with the right stuff in it, unless that cable goes to a box nearby. You won't be able to cut the cable and get it into and out of a box with any wire left to work with, unless there is an abnormal amount of slack in it. Splices must be in a box that remains accessible. So the important thing is probably where the nearest switch or receptacle box is that has a hot and neutral on a suitable circuit.

"The fixture itself is a fan/light combo where both light and fan turned on at the same time. Is it possible someone rewired the same fixture intentionally (I don't know why though) and turned it from a light-OR-fan to a light-AND-fan? I dunno, maybe they figured they never used one without the other and got tired of turning on both? Would that make more sense?" Ah hah! Yes, I'd say it is all but certain now. The light/fan functions were once separately switched and then someone decided to tie them together on one switch. That means "Plan A" because "Plan B" was based on the (now debunked) assumption that there's only one remaining function in that fan! [Not to complicate things just when we're drawing down to some final conclusions, but it also means you have available a "Plan C", which would be to use a new dual switch and return to the original design, with fan and light controlled by their own switches. Arguably, that might save some energy if the fan is sometimes left running when the light isn't needed or vice-versa. But I'm not pushing that one way or the other; it's altogether your choice!] "I will try as you suggest (requires a hardware store run) but I'd like to understand it better first. What are the fat white wires? Am I right in guessing the dark rubber conduit probably goes to the fixture and the silvery ones are household supply? Ya know, when I looked at the pic with the white wires, I didn't quite understand it either. It looks to me like they are neutrals, but I assumed you'd just chosen to omit some of details, like how the neutrals are wirenutted to each other in the box. One of the missing details about the fan/light is that we never did figure out the source of the neutral for that fixture. With 120V wiring, any light, fan or other outlet needs a hot feed and a neutral to work. These days, most electricians would wire that fan/light combo by running a three-wire cable from the switch to the fan (e.g., black for the switched hot light feed, red for the switched hot fan feed and white for the neutral). My guess is that whoever wired your light/fan didn't have 3-wire cable (or didn't understand how to use it) and so s/he put the two switch legs in one cable and ran the neutral in the other. The fact that both of those cables have rubber sheathing tends to confirm that those were part of the same "project". Why, exactly, there appear to be two white neutrals heading "north" is anyone's guess. It's rather bizzare, actually. Not that it matters much, but I think what you have there are cables (not conduit); conduit is like a pipe with individual wires running through it. If, as I suspect, it's cable, not conduit, the rubber sheathing could indicate that it is submersible pump cable or motor wire or something like that. Very possibly, it was just some scrap wire that someone happened to have laying around. Another possibility is, gulp!, that's a cut-up heavy duty extension cord! I dunno, but it isn't anything "standard" I'd expect to see in a bathroom wired in the 60s (or now either, for that matter!). [Not that I doubt you, but I just gotta ask it again: Are there really two white wires in the same two-wire cable???] The "silvery woven" cables appear to be early 60s vintage "Romex" (a brand name, actually, but often used generically). 1967 was right about the point in time when the woven sheathing was phased out in favor of the plastic-sheathed NM-B that we see today in new wiring. Whatever the ancestry or breed of that rubber-covered wire is, the way I would expect it to have been wired is this: three white (or maybe grey by now) neutrals tied together--one from the feed in the lower left, one to the circuit continuation in the lower right and one to the rubber cable in the upper right. Because a cable with two white wires is quite strange, I'd expect to find, on close examination, a black wire in that cable that's been clipped off and not used. As you've presented it in your drawing though, it's a head scratcher alright. It could be that the neutral runs on one wire to the fan and then back to the circuit continuation but, again, that'd be beyond strange, IMHO. This particular box was apparently ungrounded. The ground wires were twisted together but left loose in the box. All the other boxes I've opened up so far have had the grounding wires twisted together and looped into a tab-like projection on the box (metal), and not on the switches or receptacles. Was this acceptable practice in the past and not now? I was expecting this project to be a simple, straightforward direct replacement project. Geesh. Geesh is right! Nothing short of cucumbers grows faster than a home wiring project. Yes, times have really changed when it comes to grounding. Cables with ground wires included began to appear during the 50s and, frankly, I think a lot of electricians had no idea what to do with them! So they twisted them together and sometimes attached them to boxes, etc. :-) Keep in mind that most receptacles then accommodated only two-prong plugs and switches had no grounding terminals. A lot changed during the 60s, as three-prong outlets became standard by the end of the decade and more and more equipment had separate grounding connections. And the good news is this: Few changes in wiring requirements reduced the number of serious house fires more than the advent of comprehensive grounding systems. And the other good news is: Your house wiring includes equipment ground wires. Whenever you have the opportunity to improve something, put them to use! And, especially, whenever you're putting in a device that has a green grounding terminal, take the time to connect it to ground. (Also, check the connections on the old ground wires to make sure they're sound. Once upon a time, just twisting them together was a normal practice. Modern code requires a mechanical connector such as a wire nut or a crimped sleeve. "I really need to pick up a wiring book. Can you recommend one? I heard that one which was previously highly recommended (and I was going to pick up) is on recall now though. Yeah, that recall caused quite a flap. I'm sorry I don't have a good recommendation on the title of a basic book on the top of my head. Hopefully, someone else will pop in with one.

It sounds like one of the legs from the 3wire is providing a constant(unswitched) feed to something else at your fan box connecting to the 14/2 at the fan box. Not sure what you mean by adding another switch to make it a 3way switch. Are you wanting to add another switch at the switch box so as to power the fan and light separately? Rather then me waste time explaining something that you are not asking for, report back to that question.