Throwing exception in initialisation list?

If you cannot do the check in mtm::Dimensions, it really should be there, you can use a helper function:

int throw_if_not_positive(int x) {
    if (x <= 0) throw mtm::IllegalArgument();
    return x;
}

Game::Game(int height, int width) : 
    dimensions(throw_if_not_positive(height),
               throw_if_not_positive(width)), 
    board(height * width, nullptr) 
{
}

Or use unsigned, or use

struct positive_int {
     int value;
     positive_int(int x) : value(x) {
        if (x <= 0)  throw mtm::IllegalArgument();
     }
     operator int(){ return value; }
};

Game::Game(positive_int height, positive_int width) : 
    dimensions(height,width), 
    board(height * width, nullptr) 
{
}

You can first catch the bad_alloc thrown in the construction of board, and then throw your own custom exception:

Game::Game(int height, int width) try : dimensions(height, width), board(height * width, nullptr) 
{
    if (height <= 0 || width <= 0) {
        throw mtm::IllegalArgument();
    }
}
catch(...)  // or specifically bad_alloc
{
   throw mtm::IllegalArgument();  // or some other custom exception
}

Here's another (better) answer that's based on your own suggestion in the comments:

Game::Game(int height, int width) : 
  dimensions(height, width), 
  board((height * width > 0 ? height * width : throw mtm::IllegalArgument()), 
        nullptr) 
{
  // ...
}

Here's a demo.

Insert a helper function to validate the height and width.

size_t helper(int height, int width)
{
    if (height <= 0 || width <= 0) {
        throw mtm::IllegalArgument();
    }
    return height * width;
}

Game::Game(int height, int width) : 
    dimensions(height, width), 
    board(helper(height, width), nullptr) 
{
}