Can Mathematica Take Advantage Of a AMD Ryzen Threadripper 2990wx 32 Core Processor?

If I upgrade to a 32 core processor, will Mathematica use all 32 cores for ParallelTable or will it only use a portion of that?

It will only use as many parallel kernels as your license permits. For the student desktop versions you can look it up here. As you can see, you won't have more than 4 computational kernels.

With my license, I have 8 computational kernels and when I use the preferences to force it to start 10, I get an error message that clearly says that 2 of them could not be started

The other type of parallelism that is provided through Compile with Parallelization -> True and other built-in functions of Mathematica does not have this restriction though.

If you go to Preferences->Parallel, you should see a section that shows how many cores Mathematica can use. Off the top of my head, I don't think the Student Edition can use 32 cores, but more generally Mathematica will use as many cores as you provide it.