In order for your question to make sense, you need to define the terms "successful mathematician" and "good university". An idealistic response might skirt the issue altogether and claim that it does not matter what other people think. On the other hand, you are constantly being evaluated throughout your schooling and well into your professional career. Maybe you can focus on what it means to do good mathematics.

In his essay "What is Good Mathematics" Terrence Tao explains why he thinks Szemeredi's Theorem is good math. The theorem states that any subset of natural numbers with "positive upper density" contains arithmetic sequences of arbitrary length. Although this was proven by Szemeredi in the 1970's, it was proven by many other people using tools from all over mathematics: dynamical systems, Fourier analysis, hypergraph theory.

One of my favorite solutions is the proof of the Baik-Deift-Johansson conjecture on the longest increasing subsequence of a random permutation. Proofs of this theorem relate this statistic to eigenvalues of random Hermitean matrices and to the lengths of random Young tableaux under Plancherel measure. Again the techniques use here come from different branches of math: e.g. the Riemann-Hilbert correspondence, the representation theory of the symmetric group, orthogonal polynomials, random matrices and quantum gravity. See Longest Increasing Subsequences: From Patience Sorting to the Baik-Deift-Johansson Theorem

Good mathematics takes a certain mixed of creativity and technical know-how to pose and solve. By solving thousands of problems, you can develop your own mathematical taste. Then you can judge for yourself what good mathematics is and what it isn't.