OP
That brass is now fire-formed to your chamber, is whats happened. In my rifles, I've found it takes 3-4 firings for that to actually happen. After it does, you need to verify exactly where on that case it's become too tight for you to chamber anymore ... I say that because sometimes it isn't just shoulder bump that's needed to make them chamber reliably again.
Your 7/338 is an oddball chambering, likely with no set standard for dies like plain .308's for example have had for multiple lifetimes already. [ If ] you've checked the shoulder (the way many, many guys have suggested) and that still isn't where your problem is, use a good caliper and measure how much those cases are swelling on their diameters instead of how far they've stretched lengthwise the way everyone else has been hounding you to do. Big, fat, overbore cases like this one often don't get sized down along their length properly and there can be a portion ahead of the case-head that swells as the brass is thinned in that spot, thru repeated firings and re-sizings. Without very-good, dimensionally tight dies guys end up pulling their hair out just like you, frustrated with chambering problems. Belted mags for example, need body dies to deal with that problem because they headspace off their belts, but straight cases with no belt will do it too over time as well, because of case thinning and wall expansion.
I have brass that does exactly this around the 5th firing through my rifles, and I use a special full length die without a shoulder or neck on its top, (acting as a sort of body die like the belted guys have to use) I run the brass at that point through that special tight die to get by the problem and score a few more loadings before the brass is too-thin, and toast. It's a sign in itself that your brass is nearing the end, and often shows before primer pockets start to go. So keep that in mind too.
My bet for you, (IF the shoulder bump already suggested 15 times prior to my post hasn't cured your problem), is that your sizer is probably cut to a sloppy spec, and likely isn't sizing down the length of your brass tight-enough, far enough towards the case head. Good chance there is swelling starting a short ways up from the bottom, like I say. And is probably the root of your grief.
