Why are two planes parallel to the same line not necessarily parallel? What is a case in which the statement, "Two planes parallel to the same line are parallel" be false?
 A: Another example: take a right cylinder. Every tangent plane to the cylinder is parallel to the cylinder's axis.
A: If you take any two intersecting planes they will both be parallel to the line formed by their intersection, but they can't be parallel to each other because they intersect.

In the image above both planes are parallel to the line defined by points A and B.
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A: Because in three-dimensional space, the planes that are anchored to a line still have one more degree of freedom. They can rotate around the line. So the two planes that are parallel to the same line could be at an infinite number of angles to each other.
Making lines parallel in 3D is simpler than making planes parallel, because planes have extra dimension. You can think of a plane as an intersection of two lines.
So as you can see there is one more line involved. Omitting that extra line is what makes your statement incomplete. Not necessarily false. It may still be true in one case out of infinity.
If you had two lines crossing each other (not necessarily at the right angle) and you had two planes that were parallel to both of the lines, then you could guarantee that the two planes are parallel, because then you would have anchored both of the plane's dimensions, not just one.
A: 

Think in the floor of your room as the gray plane and think in the wall of your room as the blue plane. the blue line will be the line between the wall in front of the wall blue and the ceiling.

A: A very simple example is two different planes containing the same line: both are parallel to that line and to any other line parallel to it.
A: If both planes are infinite, and not parallel, their intersection will be a line, and any line parallel to that line will also be parallel to both planes. If the planes are parallel they of course won't have an intersection (Thanks @vlasic).
A: In $\mathbb{R}^3$, the planes $y = 1$ and $z=1$ are both parallel to the $x$-axis.
A: All the given solutions are particular case which can be generalized in the following context:
consider that in a three dimensional space, a given plane has infinite parallels planes; the order of such an infinite is one (infinite of the first order, i.e. you can fix a single parameter, for instance the distance from the line and the plane, to identify the plane).
A straight line, in the same three dimensional space, has infinite planes parallel to itself; but the order of such an infinite is two (infinite of the second order: you need two parameters (distance and angle in a given space of coordinates).
So, we simply cannot compare the two situations: the only case in which the straight line and the two planes are parallel, happens when the angle is fixed in an opportune way. So, you remove a  degree of freedom and finally obtain infinite^1 plans parallel to the line and each other.
A: The answer to Why are two planes parallel to the same line not necessarily parallel? is: because for every two planes there is a line parallel to both of them, so if the two planes parallel to the line had to be parallel, every two planes would be parallel, which implies all planes would be parallel. Which is false.
