Let's suppose I have an NP-complete problem A.
Can there be $A_1$, $A_2$ such that $A_1$ and $A_2$ are disjoint, $A = A_1 \cup A_2$, and $A_1$ and $A_2$ are NP-complete?
My guess would be yes. For example, just partition SAT into formulas with an even number of variables and formulas with an odd number.
Follow up: Can I partition $A$ into infinitely many such $A_i$? (I suppose yes: take formulas with $2^n$ variables, $3^n$, $5^n$, $7^n$, $11^n$, or something like that)