How good does my system have to be to realize an improvement with cables?

Typically, the rule of thumb is ~10%

On the other hand typically your components need to be of a significant resolution and quality before you can really expect a big diffference. I guess, I would say $2,500 components. But, some folks do realize gains with less expensive components.

These rules can be broken. I remember when my average component cost was $5K. My dealer lent me a set of interconnects costing $2K each. I bought three pair! Completely worth it. I had recently upgraded and was very much aware of how much sound quality I would have gotten by buying components that were $2K more and these cables were a screaming deal. 
 

It really pays to try and not buy. Borrow from dealer, friends, lending library. The sound improvement needs to be worth it. Many times in budget systems it is hard to tell.

My honest answer is, some people can train themselves to hear through a lot of crap and can hear a difference with mediocre systems.  I cannot, so I have to have a very good room first, then everything else follows, and then I end up caring about cables a lot less than when I started.

Cables can make a difference and your selection of cables and systems must be synergistic.

There are quality manufacturers of cables that are economical.  I used Blue Jean Cables and I preferred them to Audioquest Rocket 33s.

I didn't explore higher end cables (anything over $100) until my electronics had the resolution.  For me that was when I acquired McIntosh pre and amp.