string source = "/once/upon/a/time/";
int count = 0, n = 0;
while ((n = source.IndexOf('/', n) + 1) != 0) count++;
A variation on Richard Watson's answer, slightly faster with improving efficiency the more times the char occurs in the string, and less code!
Though I must say, without extensively testing every scenario, I did see a very significant speed improvement by using:
int count = 0;
for (int n = 0; n < source.Length; n++) if (source[n] == '/') count++;