These show up all the time in various generalized covering space theories, the reason being that you want homotopy lifting with respect to a certain class of spaces that you're interested in (e.g. locally path-connected spaces) which includes all cubes $[0,1]^n$ but probably not *all* spaces.

An important example with a simple construction is the generalized universal covering $p:\widetilde{\mathbb{H}}\to\mathbb{H}$ of the Hawaiian earring $\mathbb{H}$ that I describe in [this answer][1]. It is characterized by it's lifting property (described in link) which shows it to be a Serre fibration. It is not a Hurewicz fibration since it doesn't have homotopy lifting with respect to non-locally path-connected (even contractible) spaces like $\displaystyle\frac{[0,1]\times\{1,1/2,1/3,...,0\}}{\{0\}\times\{1,1/2,1/3,...,0\}}$.




  [1]: https://mathoverflow.net/questions/163847/are-the-higher-homotopy-groups-of-the-hawaiian-earring-trivial/163850#163850