The problem was that almost all of the rods got stuck - and it was while they were being lowered, not after.
Leading up to the accident, they had run the reactor at low power and had built up a lot of Xenon. The Xenon was poisoning the reactor, so to keep the reactor going, they needed to pull more and more rods - eventually pulling all of them or almost all of them. That was against the rules - they should have just let the reactor die down and try their experiment some other day. What happened next was that the reactor power started rising and the Xenon was rapidly being exhausted. That created a crisis where they needed to get those rods back in ASAP. As it turned out, ASAP wasn't fast enough. And the rods, almost all of them, stuck in a middle position. The problem occurred at the bottom half of the reactor that still had the moderator section of the rods - and the biggest problem was at the very bottom that was not suppose to operate with a moderator at all.