Anyone working in a facility like this is absolutely wearing multiple dosimeters.

They will have a cumulative exposure meter, this measures a total amount of dosage over a period of time and is how official things like monthly/yearly/lifetime limits for radiation workers are calculated. This meter is for things like "oops you went 2% over your yearly radiation dose and it's only November you have to stop working until January or you increase your lifetime cancer risk by .94%"

They will also have an acute meter, this is basically a tiny geiger counter which alarms above specific levels. This is for situations like "oh shit that pipe you just stepped next to has a leak and if you don't leave the room in the next 60 seconds, your never going to have any kids (or much worse)."

In some highly critical locations, it's possible for people to wear multiples of each of these, especially the cumulative meters because they are cheap and there might be multiple stakeholders who each have a process for ensuring compliance (e.g. your employer requires one, the facility requires one and the government requires one) -- this is rare bureaucratic nonsense but happens.

For the acute meters you might have multiple just for redundancy. It would really suck if you don't find out you're getting an abnormal dose until your friend walks into the room you've been standing in for an hour, or worse, when your cumulative meter is read days/weeks later.

I once had the pleasure of seeing Terry Pratchett speak, at the first Penguicon, near Detroit, MI. (Somewhen between 2006 and 2009, I think)

He told us, among other things, of working at a facility where they had to wear some sort of dosimeter, and were not allowed to leave the base without treatment if the card turned black, or something like that.

Then he mentioned one of the workers who turned up too hot to get IN to the base.

Turned out, he'd acquired some old aircraft instruments that used radium paint for nighttime visibility, and had been messing around with them.

Fun times!