Yes, it is indeed due to Capillary action. As there are great answers already by Dynadin and Aravona and Snow Crash, I would strictly limit the scope of my answer to why it happens and how, just to clarify how the Capillary Action comes in place.
The basic property of water molecules is staying together, we refer that as Cohesion, and those molecules also tend to stick to whatever that comes in contact, thats what we call Adhesion. The Physics defines Capillary action as the movement of water molecules (for layman, water as whole) within the spaces of a porous material or surface due to adhesion, cohesion, and surface tension.
Cohesive property of water is what causes to inhibit Surface Tension which is a property inhibited by liquids to acquire surface area as less as possible.
Best example to explain this would be half a glass water on a hard flat surface and you wiping it with tissue paper. Tissue paper is fibrous and hence porous.
As Dynadin explained (quoting from his answer)
In older tents the hydrostatic head was usually quite low. In addition they were quite often base untreated materials, cotton, nylon, etc. When touched there was nothing to stop capillary action from happening so the inside of the material became wet.
which clearly explains why water drips in through. The only assertion that I think everybody missed is Surface Tension that comes in picture along with the Sorptivity of the tent fabric. The technical term (when specifically considered with Capillary Action) to what we are vaguely referring as permeability or amount of leaking that can take place is Sorptivity, which wikipedia defines as:
measure of the capacity of the medium to absorb liquid by Capillarity
Mathematically (and in the harder way to look at it),
V = AS(t1/2)
where, V is volume of the water intake, A is the surface area subjected to water, S is Sorptivity of the tent surface and t is the time for which you have pitched the tent while something is touching the wall.
One question: Have you ever bought a cheap rain-jacket? Gets wet at the elbows first than belly.
Try this, take a cloth which is not really rainproof. Put it onto a vessel as a filter. Pour water quickly, faster and in more volume than what cloth can filter and drip below. You should see water being gathered, slowly filtered and then collected in the vessel below. Now, somehow, put your hand in the vessel and poke the filter-cloth from inside, water shall be collected faster than before. End of play, throw the water away to the plants.
So, What Capillary Action has got to do with something touching the tent from inside?
It has all got to do with the Porosity and the Sorptivity of the surface and the Surface Tension of the tent fabric.