Caller ID is was "Elizabeth NJ."  No message left.  Presumed telemarketer.

A new number for me to block. Called my cell, left no message

Called twice in three days; left no message. Caller ID says merely "Elizabeth."

So sick of these people ringing my phone then hanging up when they realize the machine is the only thing they are EVER going to get...

That's not in their minds. If at first they don't succeed, they're going to try try again since there's some kind of answer. If they're getting no answer at all, even with an answering device, you may be able to find your way onto their no call list since numbers with nothing are worthless.