People are telling me they are being turned down for networking meetings because there are too many other people out there asking for the same networking meetings. OK, so maybe. Let's look at this from a different angle so you can overcome this potential obstacle.
Job Search and Dating are quite closely analogous. When you were dating someone, you were "off the market", which meant that you weren't looking for a relationship. However, you did have a life. While dating, you had your own interests and spent time outside of your relationship doing things that interested you or that you had to do. While dating, did you ever meet someone who caught your interest? Did someone sweep you off your feet even though you weren't looking? How did they do it? Did they complement you, did they find common interests, were they persistent in their contact with you, did they begin to build a friend-like relationship?
People who are working are similar to those who are dating. They are "off the market". They are not looking for a new relationship, unless they are unhappy with the one they have with current employees. Even if they are looking, many people don't like blind dates, and meeting someone they don't know or who may be a referral, can be like a blind date for them. In other words, some people just won't network or don't like to unless truly compelled.
In order to find and meet the employed, you need to try your dating approach. First, identify who you want to meet? Then catch their interest by using flattery, finding common interests, persistence, building an non-threatening relationship.
To play this out further, let's say you want to meet Jim, the owner of a company. Jim doesn't have any job openings. Why would Jim want to go out on a date with you if he isn't looking? He doesn't want a date. He may however have other interests. Where does Jim go or what does he do professionally or outside of work? Does he golf? Does he belong to the Chamber of Commerce? Is he on a board? Do you have any common interests in those areas that could be used to build a non-job-seeking relationship? Or, could you share industry/work related information with Jim until he feels like you are a friend?
The point is, that in order for someone, anyone, to want to network or meet with you, there has to be something in it for them. Some people are just nice and want to help, perhaps they are empathetic. You still have to give them a reason to meet with you, especially because they are probably receiving requests to meet with others. Why you? Why are you the best date they are going to have all month?