What does it mean if data follows a t-distribution? If I make some experiment, which I repeat $N$ times, and look at the distribution of the experiment. I would usually expect (and hope) for a Gaussian or normal distribution. 
If however my data is T-distributed, what is this saying about my data? What can I interpret from the T-distributed.
I realise this may be a slightly open ended question, but if I may illustrate what I am after by means of example. If I make an experiment and my data comes out as a Rayeligh distribution, I can already say that my data is comes from the absolute magnitude of two components which are both Gaussian distributed. 
 A: If your data resembles as if sampled from a t-distribution rather than from a normal distribution, that means it has fatter tails, while it is still symmetric. You cannot say from this how the data was generated, there must be many possibilities.  But it must be some process that for some reason tends to produce many atypical values outliers.
One way to learn about this is using simulation, in R try x <- rt(100, df=5) and experiment, plot, ...
Sometimes a t-distribution is assumed to make a more robust model, see Why should we use t errors instead of normal errors?  and  Fitting t-distribution in R: scaling parameter
Your conclusion in the last paragraph is wrong, as said in comment by whuber:
Your logic is inverted: when your data are the norms of Gaussian vectors, it's reasonable to use a Rayleigh distribution to model them; but when a Rayleigh model fits your data, that does not imply your data were generated by norms of some (hidden) Gaussian vectors.
Similarly, there's very little you can say about the process by which -distributed data were generated.
