Although the title says it all, I'm not sure I'm clear enough. :)

In a dedicated box shared by many clients, what limits me first that'd make me get another box? Bandwidth, memory, processor, hard disk or other (please specify)?

I think it depends... in my case it is usually processor/bandwidth. I never got problems with disk usage or bandwidth.

There is no straight answer to that question. First it depends on your server specs. Then it depends on your target market. Then it depends on your plans.

There are so many things to figure into the full capacity of a server and which area you expect to max out first you can't really make a decision based off one either.

All the resources limit it. The first you may encounter is processor limitations or bandwidth. Some database intensive sites will take a lot out of your RAM and processor, others will barely use any. Best policy is to monitor usage on a server by server basis and upgrade or get a new server when there is a big enough load on the server.

Every server is unique. Therefore there is no real answer to your question. Its like the question, how long is a pieice of string? It all depends on the string and in your case, it all depends on your box.