A 4-bit microprocessor can do any manipulation that can be done 4 bits at a time.
All depends on how you encoded it and what it is hooked up to.
If you only use 4 bits per instruction, then you can have 32 separate instructions.
If one of the instructions is "treat the following two nibbles as one instruction" then you could, in principle, have 256 additional instructions - each taking 3 machine cycles to load. But an instruction could be "run in 8-bit mode" in which case everything takes two machine cycles to load.
It depends on how clever the designers were.
The 4004 had 46 instructions... 41 were 8 bits wide.
But you can do a lot with 32 instructions.
Before the 4004, the CPU was not even all one component.