Does $\mathsf{P/poly}$ have subexponential-size bounded-depth circuits? Is there any plausible complexity/crypto hypothesis that rules out the possibility that polynomial size circuits have subexponential-size (i.e. $2^{O(n^\epsilon)}$ with $\epsilon&lt;1$) bounded-depth ($d = O(1)$) circuits?

We know that every function computable by a $\mathsf{NC^1}$ circuit can be computed by a size $2^{O(n^\epsilon)}$ depth $d$ circuit (using AND, OR, and NOT gates, unbounded fan-in) (for every $0 &lt;\epsilon$ there is a $d$ and $d$ can be taken to be $O(1/\epsilon)$). 

The question is: 


  is there a reason that would make the existence of such circuits for general polynomial size circuits unlikely?
What you ask for should have bad consequences but I cannot think of any immediately. So I have only some pointers to what we know.

Check out Viola's On the power of small depth computation The best we know is Valiant's construction for boolean circuits: log depth linear sized circuits to depth 3 subexp circuits. (We know better for arithmetic circuits.) 
There are also some results of Beigel/Tarui on ACC begin contained in bounded depth circuits of superpoly size. I don't recall it being extended to all of $NC^1$ though.