In an algebraic topology course I'm taking we are often asked to compute the homology groups of a space $X = A \cup B$ using the Mayer-Vietoris sequence, and it happens in all of the examples I've seen so far that it's possible to do this without knowing anything about the connecting homomorphisms $\partial_{\ast}$ (say on the level of chains); we only end up needing $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ and possibly some of the inclusion maps.
My guess is that this is not a typical situation; is there a relatively simple example of a nice space $X$ and nice subspaces $A, B$ such that knowing $H_{\ast}(A), H_{\ast}(B), H_{\ast}(A \cap B)$ is not enough to compute $H_{\ast}(X)$ without knowing the specific form of the connecting homomorphisms? (For maximal relevance to the course $X, A, B, A \cap B$ should be finite simplicial complexes.)