Applying Brewer's theorem (CAP) to a single node
In my opinion, the CAP theorem is applicable intentionally to distributed systems. To quote the article "Perspectives on the CAP Theorem @ IEEE Computer'2012 by Gilbert and Lynch":


  Brewer first presented CAP in the context of a Web service implemented by a set of servers distributed over a set of geographically diverse datacenters. Clients issue requests to the service, which sends back responses.


In this context, the "network" in CAP theorem refers to communications among servers (instead of communications among clients or between clients and servers). This can be justified by the following two arguments.


To quote the above article again (with emphasis added):



  Unlike the other two requirements, partition tolerance is really a statement about the underlying system rather than the service itself: communication among the servers is unreliable, and the servers can be partitioned into multiple groups that cannot communicate with one another.



In the proof of the CAP theorem in the above article, it says,



  Consider an execution in which the servers are partitioned into two disjoint sets: $\{p_1\}$ and $\{ p_2, \ldots, p_n\}$. Some client send a read request to server $p_2$.


The proof followed is based on the communication among the two disjoint sets of servers.

To summarize, CAP theorem (again, it is my own opinion) does not consider the situation in which client nodes and server nodes are partitioned. After all, we can do almost nothing in this bad situation. In other words, CAP theorem assumes that the communications between clients and servers behave well and focuses on the theory and implementation of the distributed service itself.