Whenever I run a project some of the models are flickering, it's always the same ones and I'm not sure why...
Is there something that I have done wrong, or something I could do to fix this?

If it isnt already, try to put your camera handling code in a separate
function with its own wait loop.

Even if this ISNT managable, put "proc_mode=PROC_LATE;"(lite-c) or "proc_late();"(c-script) just before the
wait in the code that contains the camera code.

I tried this idea, it didn't solve the original problem but it did help make the camera move more smoothly, so thanks.

Those models could cause the problems. Did you try to replace them with other models?

I have had other models that have worked fine, so that could be it. But in that case, why is it these models? Is there some way to prevent this, or is it just a matter of luck?

Have the Model - Skin a alpha cannel?
then export the skin in a .bmp file and load it new.

Try opening the models in MED and run through their whole animation routines.

There may be some damaged or invalid frames in there that are making a mess of things.

Hell, it may even just be the NAMES of the animation "groups". Simple upper/lower case
inconsistancies may cause "gaps" in the animations.

How many objects are in your level? Maybe it's exceeding the limit that your Graphics Card can render and it's clipping?

I don't think that is it, as it is a very basic level, made almost entirely of cubic blocks and a few models with only a few hundred faces each.