In the beginning, when theorists where trying to explain the universe in mathematical terminologies, they ran into quite a few problems. I want to explain this history and why today we have better models to explain our universe than resorting to the big bang.
The first problem, or inconsistency if you wish, was that there was not enough time to start the universe. We began it about the size of a blood cell, with a finite radius but as small as this was, it was not small enough. This caused for us to look at the universe much smaller - because if we had not, it would not allow enough time for light to reach all four corners of the universe. However, as they universe expanded, we still had this problem they found, so enter the Inflationary model.
This allowed the universe to expand faster than light for a very short amount of time and by doing so allowed light to get where it needed to be in a homogeneous sheet of radiation. Even though this seemed to smooth out the problem, other problems still persisted.
Static Universe models actually predicted phenomena much better than the Big Bang did. A few examples was, that the microwave background temperatures actually could be explained in better terms by saying it was the limiting temperature of starlight. Would make sense - note it is not completely homogeneous either. There is aboutr 10,000th degree of error in any direction we look.
We also required many adjustable parameters to make nucleosynthesis work. Interestingly, the man who created it was Fred Hoyle, one of the biggest proponents against the Big Bang. The age of supergalaxies are also said to require an age exceeding the universe itself, two scientists who were recognized stating this is Lerner and Einstein. And the average luminosity of quasars must decrease with time in such a fine tuned way that their mean apparent brightness is the same at all redshifts, which is exceedingly unlikely.
These are just some problems with the Big Bang.