I had bags piled in my room for ages (they were in the hallway, but we couldn't keep walking around them and I wanted the hall clear for safety reasons anyway). I don't have an extra room or storage. I no longer save any "special" pieces, so I can't help you there, but I can say that if you are saving them for grandchildren make sure they are stored in a way that will make them wearable. I can't say how many friends I have whose mothers or mothers-in-law gave them their old baby clothes and they were just disgusting or so outdated they would never use them.
I recently went on an organizing binge at Wal Mart after measuring spaces in closets, including top shelves, and bought a lot of bins. I found a long narrow one that fits in the top shelf of my kids' closet and I actually managed to get ALL of their in-between clothes in it and off-season clothes. I have 2 boys, so was saving stuff. We also have an immediate hand-me-down family now (one of my husband's managers, who is raising a nephew--he is one size smaller than our youngest boy), so even if I only have three t-shirts that are clearly too small, I can wash them, fold them, and get them out of the house immediately. No storage! I highly recommend finding someone who is happy with hand-me-downs if storing clothes is an issue for you. I have started just writing our last name in the tags since we are having a third boy, but if only a fraction of the clothes come back, we've put them through at least two kids ourselves.
I also try to anticipate my kids growing out of things. If my son has four pairs of jeans and two pairs of regular pants, and then one or two pairs that "just" fit and won't for much longer, I'm more likely to get rid of the almost-fitting pants now since he's more than covered for the long haul. I try to do the same thing with my own clothes. If the fit isn't great, and you've got enough else, why bother taking up the space?