How did this ceiling exhaust fan get stuck on (and why did disconnecting this wire fix it)?

Question

I'm in the USA.

One day, the ceiling fan in a room kept going and didn't turn off when the timer switch that controls is turned off. Pressing the off button on the switch didn't turn off the fan. I presumed the switch had failed, so I went and replaced the timer switch with an identical one from Home Depot (after turning off power and verifying it was off).

Ignore the (presumably) un-related light switch that happened to be in the same box here - I unscrewed it from the box as it was in my way, but I didn't change the wiring on it.

Sadly, even after replacing the timer switch, the fan would still be on whenever the breaker was on! Here's how the wiring looked at that point.

The "before" photo wasn't great, but it was wired exactly like this, except that wire called Mystery Wire was in thn three-hole Wago on the left (where the pink line goes).

Wiring before I changed anything (The switch on the right is irrelavent - it's the light switch that got in my way. The timer switch is the one on the right):

Via some (careful) testing with my multimeter, I determined that the wire called Mystery Wire was actually hot - and given that it was in a Wago with the switched wire from time timer switch and the fan hot, the fan would always be on as it had the Mystery Wire (which was hot) regardless of if the Switched wire was hot or not. So... I capped the Mystery Wire off, and now everything works as expected (the fan shuts off when the switch is off).

After picture:

It works, but here are my questions:

1. What's the _Mystery Wire/what is it supposed to do?
2. It used to work fine until suddenly the fan was stuck on. I presume this means the Mystery Wire became hot suddenly at that point and time (and it was just not hot before), but I have no idea what would cause that

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Misc info:

The timer switch had a handful of wires pre-attached. See photo (the wire that is blue-ish in the photo is really green - sorry about the somewhat-poor photo)

When replacing it, I kept the same configuration. Here's what it is:

Switch Wire Color	What it goes to
Green	Ground
White	Neutral
Black	Hot
Red	Fan in (hot on the fan)
Red/Yellow	Nothing (capped off*)

*Directions say it is for a three-way switch, and I'm not using one here)

Also, I replaced most of the wire nuts with Wagos, although that shouldn't matter at all - I checked what happened after I switched to Wagos but didn't change the wiring and had the exact same issue.

Answer 1

The logical answer is the "mystery wire" is from a 3way switch somewhere in the room, ( or possibly in an adjacent room) that someone switched on. The fan timer probably replaced the other 3way on that circuit.

Answer 2

You could have a main floor switch that controls the fan upstairs for ventilation. So basically that red wire is switched power from that switch downstairs, and it overrides the switch in the bathroom. See if you have a switch that you don't know about. Usually near the thermostat on the main level.