Can "Tables" run out of data?

Yes, they can -- but only when the program isn't allowed to get more memory (from the system).

I'm not precisely sure what the limit is on ROBLOX, or what happens when you hit it, but it will be difficult to reach that limit: as you add more things to tables, it also gets slower to do anything to them.

For reasonable programs, you won't hit this limit. You can store an incredibly excessive amount of information and still not hit this limit.

Storing one hundred thousand words, ten times, for instance, is well, well within modern memory capabilities.

Here's a script that exhausts Lua's memory:

local x = "a";

local t = {};
while true do
        table.insert(t,x);
        x = x:lower() .. x:upper();
    wait();
    print(#t * #x / 2);
end

After reaching 3758096384 (an estimate of the amount of memory it is using), it errors and reports

22:12:16.648 - not enough memory 22:12:16.650 - Stack Begin 22:12:16.650 - Script 'Workspace.Script', Line 6 22:12:16.650 - Stack End