When I applied to graduate school the first time, for my master's, I wasn't at all sure what I was doing. I only visited one school, but that seemed to make a difference. The day after I applied (in fact, around 8:00 that morning), the dept. chair phoned me and not only verbally accepted me into the program, but also offered me funding. The other three schools I applied to (and did not visit), accepted me, but no funding was forthcoming, and they were not nearly as enthusiastic about me.
The second time, for my Ph.D., I only applied to two schools. I visited one of those, and again, that seemed to help. I didn't meet with everyone I wanted to (it turned out they were on Christmas break and I hadn't realized that), but I met with the two most important professors to my interests.
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The second time, for my Ph.D., I only applied to two schools. I visited one of those, and again, that seemed to help. I didn't meet with everyone I wanted to (it turned out they were on Christmas break and I hadn't realized that), but I met with the two most important professors to my interests.

