10, 11 dimensions is a prediction of string theory, and not an apriori statement about space time. The dimension is predicted usually under a condition of consistency, which then depends of the specifics of the theory (such as fundamental particles being strings).
For bosonic string theory, that is string theory without supersymmetry or fermions, the cancellation of the conformal anomaly requires 26 dimensions. You can also come to this conclusion by doing a "light cone" quantization, which is not manifestly lorentz invariant, and then imposing the condition that the theory needs to be lorentz invariant. This condition also requires 26 dimensions. However, despite the anomaly cancellation, this theory still predicts a tachyon. In order to remove the tachyon, one also imposes supersymmetry, which changes the calculations above so that now the required dimension is 11.
There are people trying to find a way around these conditions, in a subject called non-critical string theory (the required number of dimensions is called the critical dimension). I don't know much about these approaches.
Other theories do not predict that number of dimensions. Loop quantum gravity in principle works for any number of dimensions, although much of the work has used mathematical tricks that only work in 4 dimensions. There are also old kaluza klein models, which are models of extra dimensions. These have a fair number of problems, but they can be done in many different dimensions.
