no. cardboard seems to attract dust and bugs no matter where I store it. i've tried inside in the closet, the basement, garage and attic. the only boxes we have are for the TV and computer monitor, with the styrofoam packaging. it was nice to have when we moved and since we plan to move again in a couple of years, i kept it. I prefer to store any unused toys in big plastic tubs with lids that snap on. it seems to save space, to be able to put entire sets of toys together vs putting them all back into individual cardboard boxes.
eta: i will usually hang onto a few (ie. 1 or 2 or 3) of the smaller amazon boxes as I frequently mail things to cousins or my parents, but those don't sit for long.
loving my dh and our littles (dd ~ 11 yrs, ds ~ 9 yrs, ds ~ 7 yrs, dd ~ 5 yrs and baby brother ~ 20 mo)