Is it the walls letting in the water or the roof absorbing the wet and it running down the walls?
The roof may need new felt - that's a pretty straightforward job http://www.youtube.com/watch?v=1mo5PyRRZm4
If you're sure that the rain is coming through the walls, check that they're the right way up - with tongue and groove, the tongue should be at the top.
Has the wood been treated in any way?
Hi, Jackie......I've had this problem in the past and to be honest most sheds with ship-lap or tongue & grooved boards have a little seepage during very heavy rain. However, you could seal the gaps (during a dry spell) with either clear silicone sealant or weatherproof caulking......both applied with a mastic gun.
Just another thought - was the shed under cover when you bought it? If it had been stored in the dry the wood might have shrunk and loosened the joints. If that's the case a good soaking in a few downpours should swell the joints and cure the problem
Perhaps it would be apt to add this to this thread.
The cheap mineral felt sold & used for shed roofs has a very limited life.....it usually succumbs to high winds or rotting after a very short while.
I have a large/ish shed and summerhouse, both were felted as above when new. When they needed re-felting, I decided to have them both 'hot felted' by professional roofers...it cost £100 per building and came with a 25 year guarantee.
The process involves the roof being stripped clean, an under-felt being applied, this is heated & melted with a flame-gun, next a heavy duty mineral felt is laid on to this and both are fused together, again with the flame-gun.
No nails are used and it's a really neat job.
Hope it resolves itself without too much dampness. If it continues it's possible to put an outer cladding of weatherboard on the walls - you could even introduce some insulation when doing it - just think how cosy that'd be