In dimensions n at least 4, S^2 x S^(n-2) is not cancellable, so connected sum decomposition is not unique; for simply connected manifolds the same failure is claimed for n at least 17.
On CW-complexes over groups with periodic cohomology
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abstract
If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincar\'{e} $3$-complex $X$ with $\pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $\mathbb{Z} G$ modules where $G$ has periodic cohomology.
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Connected sum decompositions of high-dimensional manifolds
In dimensions n at least 4, S^2 x S^(n-2) is not cancellable, so connected sum decomposition is not unique; for simply connected manifolds the same failure is claimed for n at least 17.