Whilst undoubtedly some individual players will raise their game when they play against any of the "big clubs" I don't think that explains the issues we have faced recently.
In terms of general attacking play all premier league clubs can now afford to buy good attacking talent from around the world. The days of the bottom half of the table just having big plodding target men only are long gone, but that applies to everyone so I don't think that's a key factor.
Under Ferguson an opposition goalie might have the "game of their lives" and hold United to a draw or even hold onto an unlikely win. On those occasions though they would have been battered from all angles, with attacks coming from multiple players, being exposed to crosses, free kicks, long balls, intricate triangles on the edge of the box, defenders making an extra man...etc etc. They would probably have had to pull off 9 or 10 critical saves / last ditch defending from their teammates.
Unfortunately under LVG and especially Mourinho we have become passive, plodding and predictable. For a lesser teams goalie to have the "game of their lives" currently they might need to pull off 4 or 5 key saves. We will probably miss 1 or 2 clear chances and we certainly won't try and overwhelm them with numbers in their penalty area.
I'm not one for searching for stats but I'm sure that the number and intensity of United's attacks has fallen dramatically and we don't sustain pressure on the smaller clubs in the same way.
So it's no surprise if we don't apply the same mounting pressure, if we don't pepper their goals with attempts from a variety of angles that opposition teams only have to raise their game by a much smaller margin than they would have had to do in the past to get a result from us.