Is it possible for two planes to meet in a point instead of in a line?
No. The only possiblities are:
1) Do not intersect. (the planes are parallel)
2) Intersect in a line.
3) Intersect in all points. (They are actually the same plane)
I have to disagree with HallsofIvy. The answer depends on the dimension of the ambient space in which the planes live.
In 3 dimensions, Halls is correct.
In 4 dimensions and higher, two planes (i.e., 2-dimensional objects) can certainly intersect in a point.
Are these linear objects you are talking about or some other kind?
Separate names with a comma.