- n ≥ 5: Smale, 1960 (and Stallings at around the same time)
- n = 4: Freedman, 1980
- n = 3: Perelman, early 2000s (this is the Poincare conjecture)
- n = 2: "follows from the Riemann mapping theorem"
- n = 1: "a nice exercise for an undergraduate"
So in low dimensions the problem is easy, or at least doesn't require "modern" apparatus; in high dimensions it's harder (Smale and Freedman both got Fields Medals); in "middle" dimensions (like 3) it's the hardest, or at least took the longest. It's my understanding that it's pretty typical in geometry/topology for the three- or four-dimensional cases of a problem to be the most difficult?
Can you think of other problems (from any area of mathematics) that have a similar property -- that they're hardest for some medium-sized value of whatever a natural parameter for the problem is? (Yes, it's a vague question.)