What are legitimate connections? Are "a lot of my friends from college moved here, I love the sports, and my aunt and uncle live in the area" legitimate enough if you've never actually lived in the city/state?
Depends on the city. For NY, pretty much saying "Because it is NY" is good enough. People there understand it is a very desirable legal market, and you don't need to prove you want to be there/stay there. For Chicago, I found you need a reason "why Chicago," but it could be minimal. The fact that I went to a midwestern school for UG and now have friends that live there was enough. Boston & DC are a little stricter than that. They want to hear that you have close family there (not your 3rd cousin once removed), you lived there in the past, you got law school in the city, etc. These cities are more concerned with people leaving the city, and want to make sure you are going to stay (Boston laweyrs-->NYC and DC is very transient). Besides those cities, I don't have much insight.
Also, it all depends on your grades. If you are in the top of your class at a good school, they care less and less about your connections. It becomes more about whether you are really interested in their city, or if it is a back up city. In other words, they will be more concerned about how many cities you applied for jobs in, how many firms in each city, where you got offers already, etc. and less so on your ties to the area (though to some extent, since this is another factor in determining what city is the applicants true 1st choice)