I went to Sri Lanka in June and was a bit worried what the weather would look like. After all, this is the intermonsoon season and you don't want your trip spoiled by gray skies and torrential rain. As many other travelers put the same question, here is some anecdotal evidence of what you can expect. Judging from my 18 days I'd say: worry not! I had sunshine almost everywhere, including:

East coast Nilaveli Beach - perpetual sunshine, as was to be expected. This is actually the East's dry season, which was one of the reasons for me to go there. Excellent swimming.

Cultural triangle (Anuradhapura / Mihintale / Pollonaruwa / Sigiriya / Dambulla) - sunshine every day with some clouds thrown in, very pleasant weather. No rain at all.

Hill country (Kandy / Haputale / Ella) - again very good weather, sun and some clouds, no rain.

Yala National Park and South coast (Mirissa / Galle) - 2 days out of 7 it was raining, otherwise very nice weather with a few clouds. Nice swimming in excellent high waves was possible at Mirissa Beach (conditions change, observe rip currents carefully before you venture into the water).

Colombo - mostly grey skies and some rain, less sunshine than at the other places.

All in all, the weather was ideal for traveling and taking photos. The first drop of rain I had only after 10 days, on the South coast. Locals say that weather patterns are becoming more and more unpredictable, you can have rain when it should be dry and vice versa. If you consider to visit SL off-season - go ahead!