Does every test statistic fit a certain distribution? Pretty much all of the test statistics fit a certain distribution. Is there a possibility that when doing my own hypothesis test, I could get a test statistic that doesn't fit any distribution?
 A: A test statistic is a random variable itself (indexed by the sample size), then it has a distribution by definition. Whether it is a well-known distribution, that is most likely not the case, but there are bootstraping techniques to approximate its distribution.
A: Under the null hypothesis and given the assumptions, if the test statistic is completely determined, it has some distribution (though it may not have a name).
There can be circumstances where it's not possible to determine the distribution of the test statistic under the null hypothesis.
For example, if we have a random sample from a normal distribution whose mean is that specified under the null hypothesis, we know that the one sample t-test statistic has a t-distribution. 
But what about if one of the following happens for example:


*

*the population we sample doesn't have a normal distribution 

*the observations have serial correlation ($\text{Cor}(Y_t,Y_{t-1})=\rho$)

*a quarter of the population has a larger variance than the rest of it
If we don't know what the distribution is in the first case, or the correlation in the second case or the ratio of variances in the third case, then the distribution function is not uniquely determined -- it depends on the thing that's not known - change that and you have a different distribution.
On the other hand, if we did know those things, we could determine the distribution. In some cases we might be able to compute it algebraically, but otherwise we could get it via simulation. For example, what if our sample size was 5 and our data were drawn from an exponential distribution? If the algebra proves too difficult, we could simulate samples of size 5 from an exponential distribution, calculate the t-test statistic and then we'd know what its distribution was:

On the other hand, if we actually know that it's exponentially distributed, a t-test isn't the best possible test for the population mean.
Even when we don't know the distribution the data are drawn from, we still may be able to do something. For example, we can use resampling, such as a permutation test.
