I would just like to throw in some more perspective here, although your main question has been answered.
First, let's see a postulated mechanism of formation of Grignard reagent in the presence of Mg and diethyl ether.
The tentative problems that would may arise by the presence of the N-H bond would be:
1)The R radical made in the first step might begin a homolysis reaction with the H of the N-H, and thereby forming a kind of amino radical radical. This may lead to further erroneous coupling in the synthesis.
2) Like the answer and comments above state, the major reason for the failure of this synthesis is the reaction of the highly basic Grignard reagent with the acidic H after formation.
3) Non-polar solvents like diethyl ether stabilize the Grignard reagent formed like so:
The N-H bond will have significant polarity due to difference in electronegativities of N and H, so a lot of ether will be expended in stabilizing this bond, so again, it might hinder the formation of the Grignard reagent
Note: This answer is more like food-for-thought on this issue as far as point 1 and 3 are considered, the clear-cut answer is in point 2.
P.S. As pointed out in the commnts by Mithoron, the reaction happens in a heterogenous state, so the mechanism cited above and all these arguments are highly simplified ways of looking at a picture perhaps not completely understood. This reaction probably bears more similarity with that of sodium and an alcohol