In 1 word, the problem is depth. From there, you can run down the list of problems that stem from a lack of depth.
The Blueline - The D isn't good enough/deep enough to hold other teams at bay. Chara was one of the best D-men of his generation. His mere presence changed the way teams would approach the Bruins' zone. You can make the argument that he impacted a game more than any player in the league, in his prime. At this point, Chara is probably a #2 guy. After him, you have a bunch of guys slotting in at least one spot above where they should be. The lack of depth can be masked for stretches of a game or season but mistakes and poor positioning will get exposed by good teams.
Forwards - This is a 1-line team right now. The second line shows flashes but isn't anywhere near as consistent as they need to be. I don't thinkI need to go into the bottom 6. I just hope they don't screw up too badly.
Goaltending - There are 2 clearly defined groups on this forum. The 'Rask Sucks' group vs the 'Rask gets hung out to dry' group. I think the answer is somewhere in the middle. Given a good team in front of him, Rask is great. Hear me out before you say "duh, no kidding". Having a good D/forward group drops Rask's workload tremendously. I watch the guy make great saves and give up a stinker or two in the same game. I think he's both mentally and physically fatigued. If he can have a good team in front of him, he'll get back to superstar goaltender status. He still has it in him to be the guy who allowed 2 goals in a series vs the Pens. I know some people will come back with, "Carey Price doesn't have a good team but he's still great". Well, ok, he's Carey Price. He's the best goalie in the league. That's like comparing Player X to Sidney Crosby. How many goalies in the league are doing Price-like things with mediocre teams?
The 09-14 Bruins were a machine who could roll 4 lines and 3 D pairs. After Savvy went down, they didn't have guys that you would consider top scoring threats (which may have cost them a series or two....Washington) but they were able to grind teams down over and over. Not just physically, but with scoring opportunities and well as stifling team D. These guys just don't have the horses. The thing that makes it so shocking is that it happened so fast. They went from President's trophy to DNQ. Now, you can blame the lack of depth on scouts, GM, coaching, or whoever you want. From an on-ice standpoin, they don't have enough good players.