Yes, every division is a noticeable increase in skill. A person who isn't analytical at all may just say "oh yeah, it's all the same thing until you get to high plat" or something stupid, but the differences are pretty clear. Sure, a silver 5 guy may not get completely stomped by a silver 3 or anything, but if you are silver 4 with 600 games played and cannot seem to find a way out, then there is a clear reason why that has happened.
I don't mean this to be an asshole, but really, just take the time to analyze everything you do. I don't play a whole lot of ranked, but every time I do I end up doing retarded plays or decisions that make me lose games in silver when I really shouldn't, but once I start doing that, I find myself catching bad calls before I make them and end up climbing again. Having mechanical skill and knowing the basic fundamentals of the game like CSing, trading, and objective priority is important, but actually understanding what you're doing and why you're doing it will carry you much further.