We establish sufficient conditions for $n$-fold bounded differentiability ("$b$-differentiability") of mappings of locally convex spaces and sufficient conditions for $n$-fold Hyers-Lang differentiability ("$HL$-differentiability") of mappings of pseudotopological linear spaces. We describe a class of locally convex spaces on which there exist everywhere infinitely $b$-differentiable real functions which are not everywhere continuous (and so are not everywhere $HL$-differentiable). Our results show, in particular, that for a wide class of locally convex spaces a significant number of the known definitions of $C^\infty$-mappings fall into two classes of equivalent definitions.