jmc

The number of degrees in a hexagon is $(6-2)\cdot 180=720$ degrees. Setting the degree of the smallest angle to be $x$, and the increment to be $d$, we get that the sum of all of the degrees is $x+x+d+x+2d+x+3d+x+4d+x+5d=6x+15d=720$. We want $15d$ to be even so that adding it to an even number $6x$ would produce an even number $720$. Therefore, $d$ must be even. The largest angle we can have must be less than $150$, so we try even values for $d$ until we get an angle that's greater or equal to $150$. Similarly, we can conclude that $x$ must be a multiple of 5.

The largest angle is $x+5d.$ We notice that, if we divide both sides of $6x+15d=720$ by 3, we get $2x+5d=240.$ For $x+5d<150,$ we must have $x>90.$ The largest value of $d$ occurs when $x=95$ and $5d=240-2x=240-2\cdot 95=240-190=50,$ or $d=10.$

Therefore, there are $\boxed{5}$ values for $d$: $2,4,6,8,$ and $10$.