When you play the Game of Thrones...

[hamlet]

Yeah, but given more or less late medieval technology, how long can you really store grain? Ten years sounds too long, and apparently, there were longer winters before.

Storing grain is actually not that difficult. Granted, in a damp climate like northern Europe (i.e., large tracts of Westeros) you can do so for a very long time indeed. Assuming you can keep it dry and relatively free of vermin, you can store good dried grain for years.

EDIT: Went and looked it up just to be sure. When the wheat is milled (i.e., ground into flour) it almost immediately begins to oxidise and "go bad" and must be used relatively quickly (within a year). Unmilled wheat berries (i.e., dried from the stalk) are virtually immortal as long as they are dried and kept in tightly sealed (preferably air tight) containers and away from vermin intrusion.

Some wheat found in the Pyramids at Giza was actually still edible after 5500 years. Brave, brave man who empirically tested that out.