My husband and I walked the Norte in sections, so it took longer than it would for a thru-hike. Still, the Norte is a bit more than 500 miles and more difficult than the Frances, so I wouldn't be able to do it in four weeks (but then I couldn't do the Frances in three). Are you used to these kinds of miles? I liked the weather--in part because it was on the cooler side, which I prefer to heat, but it was often drizzly and foggy. The forecasts, as any of the local will tell you, are unpredictable. Forecasts will say rain for day after day, but in reality it might be fog and drizzle, off and on. As said in an earlier comment, with good rain gear you should be fine. I recommend a lightweight hiking umbrella. Accommodations are spotty, but bookings.com can be your good friends. I absolutely loved the Norte (wrote a book about it) and highly recommend it, just not sure about your time constraints. You could start out from Irun and see how your pace and experience goes, then take a bus/taxi/blablacar through some of the interior, and resume your walk at Vilalba or elsewhere and walk into Santiago.