The basic, one-operator, input-output plant model y = Gu has fundamental limitations in feedback and closed-loop modelling problems on the full time axis when G is an unbounded linear operator. It has been a deeply rooted misunderstanding that there is no essential difference between singly-infinite and doubly-infinite time axis formulations of feedback problems using this basic plant model. We present here several new types of results on the limitations of the one-operator plant model y = Gu in discrete-time feedback and modelling problems on the full time axis Z. A detailed analysis of existence of solutions in two concrete feedback problems is also provided. We discuss also the errors in the classical transform domain analysis of the unbounded plant mapping case on the full time axis. An important conclusion of the present and related earlier work is that the two-operator plant model Ay = Bu should be seen as the fundamental input-output plant model. This model allows the modeling of open-loop unstable plants via two bounded operators. In this way the technical difficulties associated with unbounded operators can be avoided on the full time axis.