Implement template type constructors only if underlying type has those constructors

Is there a way to encapsulate a type in a template class (akin to something like std::optional) that has all the necessary special constructors and assignment operators (i.e. copy ctor/assignment, move ctor/assignment) but only "enables" them if the underlying type has those functions? The functions in type_traits like std::is_copy_constructible look like they might help but I'm not sure how to use them to achieve this goal. For reference, the type I'm trying to implement is similar to std::optional but instead of the alternate value being simply "none" I want to use a custom error type. E.g.

template <typename T>
class ErrorOr {
 public:
  enum class Error {
    FATAL,
    WARNING,
    NONE,
  };

  ErrorOr(T val) : val(val), error(Error::NONE) {}
  ErrorOr(Error error) : error(error) {}

  // TODO: Implement copy/move ctors/assignment operators that only
  // exist if they do for the underlying T

  T get() { val; }

 private:
  T val;
  Error error;
};

This is a very bare-bones/minimal implementation that doesn't have a lot of the necessary features but hopefully illustrates the point I'm trying to make.

Is this possible in C++11?