I think that Dave is correct - I always check the fare conditions. For a Saver fare that you can't upgrade with miles until check in, it says at the bottom of the fare conditions "NON-END/SKYWARDS SAVER/ - AND - CHK-IN REWARD-UPGRADE", whereas for a saver fare that you can upgrade with miles at any time (subject to availability!), it just says "NON-END/SKYWARDS SAVER/".

Unfortunately, I think that they've introduced Non-upgradable Saver fares in combination with Flex fares, and since the most restrictive fare conditions apply to the whole journey, it's possible that the flex leg of the trip wouldn't be able to be upgraded prior to check in either... but I don't know.