It's hard to say for sure without pulling it apart but your pictures suggest to me that you have water getting behind the wood at that corner. The reason I think this might be the case is that you basically have no visible problem except at the bottom of the board where it intersects with a horizontal piece.
What I mean to say here is that this looks to me in picture 1 that the rotted section is where the water is exiting, not entering. Contrary to popular opinion, wood is not likely to just rot from water splashing on it. It takes a while for water to penetrate wood. Whenever I've had serious rotting issues like I see here, it's when water is trapped at some choke-point within the structure. The neighboring sections look perfect and it doesn't look like there's any significant reason less rain hits those areas.
Anytime you have a water issue, you need to make sure the roof, fascia, drip-edge, and gutters (if you have them) are all super tight.
UPDATE: Because you live in a cold area with snow, I suggest you watch the roof closely during the winter, especially in that area. An otherwise perfect roof can leak if you have an ice dam. In a nutshell, you can end up with a pool of water standing on your roof. In your area, I would expect that the roof would have some protection against this (e.g. ice and water shield) but this is typically only installed a few feet up the roof. The pitch on your roof appears to be pretty gradual which means any pooling will extend farther back and it will not shed snow as readily as a more steeply pitched roof. If you get major 'killer' icicles, it's a sign you might have an issue. Ice dams are really hard to get rid of once they form and it's freezing out so the key is to remove the snow before they form using a roof rake or roof razor.
You need to pull down the wood around that corner and make sure there is no rot in the structure. You really don't want the corner of the building sagging. If find any rot in the structure, you need to remove it and replace it with new wood. Open it up. It's scary at first but you really need to do it.
I would also recommend not painting exterior wood and use a solid stain (or at least use a 'breathable' paint) instead. Paint can end up contributing to water being trapped within wood. In theory, it should keep it out but it's never completely sealed on all edges. This means moisture in the wood can be blocked from exiting.