I'm in Klamath Falls, Oregon, which is listed as zone 6, but we hit -13 this week, so maybe it's more like zone 5. We average 106-120 frost free days, depending on which source you trust. I'm new to the area this August, but according to the locals there was a hard frost the third week of June this year, so there were almost no apples or other standard fruit in town this year (but
native Klamath plums, elderberries, bitter/choke cherries, ribes species, etc. all had plenty of fruit on them).
I found a chestnut in a public park here. I don't know what species/varietal it is, but it is definitely a true chestnut, medium sized single-trunk tree. It opened it's burrs and dropped nuts at least a month after the first frost, sometime in October; however, the nuts were empty. From my understanding a chestnut won't set seed without another individual for pollination, and this one is all by itself, so that is likely the cause, though I can't conclusively state such.
To echo Adam, the large diurnal temperature swing in the arid West is fairly problematic for many species (in summer the average swing is 35 degrees here), but the chestnut I'm referring to is a fairly large healthy tree. I should state that it is within a few hundred yards of a large lake, which might moderate the climate a bit, and it is in a public park next to grass, so it gets
irrigation. As to soil, I haven't dug in that location, but based on the county soil survey, it is a loam made from sanstone, breccia, diatomite, and basalt, though I don't know the pH. The
land is slightly sloping to the north, but it isn't shaded at all, being in a open grassy area, except maybe in mid winter the sun may dip a bit below the hills to the south of it. We get just shy of 2000 degree heating days here (warm sunny summers). Just trying to give as much info as I can hoping some of it may be of use to someone. My thinking is that chestnuts could work here, based on all this, but until I see full, tasty chestnuts falling from a tree here, I can't be sure.