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Can it happen that a direct sum of a free module and a non-free module is (isomorphic to) a free module?

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    Yes. Google for "stably free module" (e.g. on Keith Conrad's page). – 2017-01-28
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    Don't you need a torsion-free condition on the non-free side? – 2017-01-28
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    What is an example? – 2017-01-28

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An example from geometry. The module over $C^{\infty} (S^2)$ of tangent vector fields on the 2 sphere is not free (every vector field has a zero), but if you stabilize it by a free module of rank 1 ( you think of normal vector field to the sphere) you obtain a free rank 3 module : the restriction of vector fields on $R^3$ to the sphere.

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    http://math.stackexchange.com/a/58035/121097 – 2017-01-29