This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following situation: find a short exact sequence $$ 0 \to X_1 \to X_2 \to X_3 \to 0$$ (in some category of your choice), and a second exact sequence $$ 0 \to Y_1 \to Y_2 \to Y_3 \to 0$$ such that there are isomorphisms $X_n \cong Y_n$ for all $n$, BUT in such a way that there is no commutative diagram whatsoever between the two sequences, with the vertical maps being isomorphisms.
It is impossible to find such an example in the category of vector spaces, or of finitely-generated abelian groups. I don't know about the general case, though.
I would be grateful for any example, but would be disappointed if the chosen category were constructed specifically to answer the problem.
EDIT / COMMENT: in the first version of this question I was asking for sequences which could potentially be infinite. Some great examples came in the comments straightaway. I'm very thankful for them, but I recall only now that the student's original question was about short exact sequences, so I've edited accordingly. (I'm sorry for the confusion, the student asked me this question several months ago, and I was posting only now, for some reason... and got it wrong. I'm very happy to know about the examples involving infinite sequences, though.)