Does field line concept explain electric field due to dipole? Consider an electric dipole consisting of charges $-q$ and $+q$, separated by a distance $2a$ and placed in free space. Let $P$  be point on the line joining the two charges (axial line) at a distance $r$ from the centre of $O$ of the dipole. 

You  can observe in the above figure that electric field has two directions at the same point $P$, does this mean electric field lines of two charges can intersect?
This is the common figure given in all text books, what I can observe is that field line due to $+q$ charge ends at $-q$ charge and then it doesn't progress towards the point $P$, and nowhere I see the lines of $+q$ and $-q$ intersecting, now I can't conclude that field lines do intersect. So, does field line concept explain electric field due to electric dipole?

What I observed in Wikipedia dipole page is that, axial line is only vanishing! 

Sometimes I might have misunderstood the concept, if so pardon me and explain. 
 A: If you take a permanent magnet, and place a sheet of paper over it.  Now sprinkle iron filings on it, and you pretty much get this diagram.  This has been the mainstay of field theory since Faraday's time.
A test charge at rest will begin to move in the direction of the field line.  Since there is nowhere that it can rest where there is more than one possible direction of motion, there must be no crossings of the field line.  
The line that disappears to infinity in one way, and reappears from the other side, means simply that the flux is moving on ever-large circles, and that in the axis-line of the dipole, it is feeding flux as a stream through it.  But all this means is that it is turning something that is already there, but never getting a full rotation of the disk up.  
In the real world, these polar flux lines simply wander off to another electrical system. Gauss's flux law says that there is a sphere with a net flux across it equal to the enclosed charge: a net of zero does not mean everywhere zero.  
A: At any point the electric field is the vector sum of the fields from the two charges. So while the fields from $A$ and $B$ are indeed in opposite directions at your point $p$ you just add them (well, subtract their magnitudes since they're in opposite directions) and this gives you the net field.
I wouldn't take the field lines too seriously. They are not physical objects, they are just notional paths following the direction the field vector points in. If you look at the length of a field line as a function of its angle to the axis you'll find the length goes to infinity as the angle goes to zero. So the field line exactly on the axis has an infinite length and therefore never reaches the other charge. But, as I say, these field lines just show the direction of the field so there's no special physical significance to the infinite length.
See also the question: Are the axial electric field lines of a dipole the only ones that extend to infinity?
