  Every theory of information, including classical and quantum, can be studied
in the framework of operational probabilistic theories--where the notion of
test generalizes that of quantum instrument, namely a collection of quantum
operations summing to a channel, and simple rules are given for the composition
of tests in parallel and in sequence. Here we study the notion of compatibility
for tests of an operational probabilistic theory. Following the quantum
literature, we first introduce the notion of strong compatibility, and then we
illustrate its ultimate relaxation, that we deem weak compatibility. It is
shown that the two notions coincide in the case of observation tests--which are
the counterpart of quantum POVMs--while there exist weakly compatible channels
that are not strongly compatible. We prove necessary and sufficient conditions
for a theory to exhibit incompatible tests. We show that a theory admits of
incompatible tests if and only if some information cannot be extracted without
disturbance.
