We consider the superposition of macroscopically distinguishable states for a measuring process whose time evolution is described by the Schrödinger equation. We ask whether it is possible to observe interference effects due to the above mentioned superposition and how to observe them, taking into consideration an experiment performed by other authors. We find a necessary condition in order to be able to observe these effects. We also point out some very serious difficulties in observing them and analyse the connection between (...) some of these difficulties and the Wigner-Araki-Yanase theory. We try to explain why the whole problem seems to us to be far from having a solution and we suggest some paths we think worthy of further investigation. (shrink)