C++ - could construction of a non-trivial pointer supplied by an allocator type throw an exception?

This question is not limited to the built-in C++11 pointer types (shared_ptr etc) but includes any custom pointer type that could be potentially defined in C++ and included as part of a standards-compliant allocator.

Could the construction of a non-trivial pointer, such as that supplied by a custom allocator (std::allocator_traits<Alloc>::pointer) throw an exception and if so, why?