Let $E,F\in D^b(X)$, where $D^b(X)$ denotes the derived category of coherent sheaves on some smooth variety $X$. I am thinking about the following question:
If $E \oplus E[1] \simeq F \oplus F[1]$, is it true that $E \simeq F$?
One can also ask the similar question where 'shifted by $1$' replaced by 'shifted by $n$' for any fixed $n\in \mathbb Z$.
Take any $X$ which has a vector bundle which is stably free but not free; say there is a vector bundle $A$ has rank $n$ that is not free but $A\oplus\mathcal{O}_X$ is free. Then consider $E=\mathcal{O}_X\oplus A[1]\oplus \mathcal{O}_X[2]$ and $F=\mathcal{O}_X\oplus \mathcal{O}_X^n[1]\oplus \mathcal{O}_X[2]$. Then $E\oplus E[1]\cong F\oplus F[1]\cong \mathcal{O}_X\oplus\mathcal{O}_X^{n+1}[1]\oplus \mathcal{O}_X^{n+1}[2]\oplus \mathcal{O}_X[3]$ but $E$ and $F$ are not isomorphic. (A similar example would work for shifts by an arbitrary nonzero integer; just multiply all the shifts here by that integer.)