Recent findings show that single, non-neuronal cells are also able to learn signalling responses developing cellular memory. In cellular learning nodes of signalling networks strengthen their interactions e.g. by the conformational memory of intrinsically disordered proteins, protein translocation, miRNAs, lncRNAs, chromatin memory and signalling cascades. This can be described by a generalized, unicellular Hebbian learning process, where those signalling connections, which participate in learning, become stronger. Here we review those scenarios, where cellular signalling is not only repeated in a few times (when learning occurs), but becomes too frequent, too large, or too complex and overloads the cell. This leads to desensitisation of signalling networks by decoupling signalling components, receptor internalization, and consequent downregulation. These molecular processes are examples of anti-Hebbian learning and 'forgetting' of signalling networks. Stress can be perceived as signalling overload inducing the desensitisation of signalling pathways. Ageing occurs by the summative effects of cumulative stress downregulating signalling. We propose that cellular learning desensitisation, stress and ageing may be placed along the same axis of more and more intensive (prolonged or repeated) signalling. We discuss how cells might discriminate between repeated and unexpected signals, and highlight the Hebbian and anti-Hebbian mechanisms behind the fold-change detection in the NF-κB signalling pathway. We list drug design methods using Hebbian learning (such as chemically-induced proximity) and clinical treatment modalities inducing (cancer, drug allergies) desensitisation or avoiding drug-induced desensitisation. A better discrimination between cellular learning, desensitisation and stress may open novel directions in drug design, e.g. helping to overcome drug resistance.