Yes. This is possible. But if you are "stuck" in a league, don't get your ego up thinking its your MMR causing problems.
MMR is invisible, like you said, but it can still be observed at the end of games. LP, the thing we can see, is actually less important (Ironic right?).
MMR determines who your opponents will be, but there are average MMR values for each league that occur naturally. So lets say you get placed into silver 3. You queue up for a ranked game and win it. Your MMR goes up, and you get 16 lp. You queue up again, win again, and this time get 20 lp. I'm speeding up this process a bit, but in your third game you get 25 lp for the win (HOT STREAK).
What is happening? As you win, you are gaining lp, but you are also gaining MMR. What is going on is the game is seeing that your MMR is higher than the opponents in the league you are in, and starts giving you more LP per win so you rise through the divisions and leagues faster to get to your skill level. So consistently winning games will cause you to earn more lp per game because the game thinks you are much more skilled than the people you are playing against. The same is true in the opposite direction as well, go on loosing streaks and your MMR will start to fall, resulting in the game thinking you are in a league out of your league (see what I did there?) and you will start to lose more lp per win. There is a buffer of a few games before MMR starts to fall (and lp changes become dramatic), but in general thats how it works.
So if your MMR and rank are initially below where you should be, as you win games (because you are a better player than your peers in the division) your MMR will rise. And then your MMR and rank will bounce back and forth as you get closer and closer to your true MMR.