"The difference between theory and practice is that, in theory, there's no difference, but in practice, there is."
At Colorado State University over the past several years, students using the Teachspin muon lifetime apparatus have been consistently getting muon lifetimes that are slightly but significantly too short. In late 2012 I spent several months collecting my own data, and re-analyzing the student data from spring 2012. My results are explained in http://www.riverrock.org/~howard/Teachspin_error.pdf (Ignore the (relatively small amount of) material about QTD and Van de Graaff generators; they are irrelevant for your purposes.) There are many possible sources of experimental error, some of which I detail in section 4. I also explain how to correctly analyze a truncated exponential (or geometric) distribution, and give source code for all my analysis software.
Section 3.6 explains that "a property unique to the exponential and geometric distributions is that they are memoryless; the probability of an event occurring does not depend on how long we have already waited for it to occur. Another way of saying this is that they have a constant failure rate. Because muon decay (like nuclear decay) follows an exponential decay pattern, we do not need to be concerned with the history of the muon before it arrives in our apparatus. Indeed, the entire experiment would not be possible without this property, since we do not measure the entire lifetime of the muon from its creation in a pion decay while still traveling at nearly lightspeed, but only its lifetime after coming to rest in our detector." These distributions are also the maximum-Shannon-entropy distributions given only the constraint of having a particular mean lifetime, which means that assuming any other distribution of decays is equivalent to claiming that you know something special about the internal decay process, that allows you to predict it better.
After the above paper, I was able to demonstrate that a part of the error was due to contamination by shorter-lived particles like pions (26 nS) and K_longs (51 nS). Since CSU is at about 1500 m altitude, a few more of these should be expected than at sea level, but the data indicates even more than that. The best guess at the moment is that the excess were "structural" pions generated by cosmic rays colliding with the steel and concrete of the physics building; these wouldn't have to travel very far. They show up as an increased number of events in the first few decay bins.
However, pion contamination is insufficient to explain the whole discrepancy. Much of the remainder is still a mystery.