Recently the toilet tank has begun not filling up when flushed until another faucet is opened elsewhere in the house.

Sounds strange. What type of fill valve do you have in you tank? Does it have a rod with a ball type float on the end, or does it have cylinder looking float?

Im betting on a faulty fill valve, but you say it will fill when another faucet in the house is turned on. That the confusing part. Definitely have me curious. No matter what you find, please let me know.

It's my bet that your ballcock's a float ball type, (see image).

Try this. With the tank filled take the lid off. Have someone watch the float ball while you turn on the nearest colld water faucet. Now shut it off fast. Did the float ball bob up and down when you shut the cold water off? Let me know, Yom