Weather changes abruptly in higher altitudes. Normally the rain starts in the mid or end of June and lasts till mid-late September. Yes, U can go for the 1st week June, but do keep necessary accessories to keep the rains away, if it may occur. Do keep a check on weather forecast. Also, October onwards till Mid-December is a good time to visit SIKKIM. December to March - risk of road blockages, due to snowfall, in NORTH & EAST SIKKIM might spoil the fun.
First thing is we are not copying paste the google things we have visited there thats what we are here to share with fellow travellers. As mostly In march to May visited so we went in May end and great time with some rain in Sikkim , Gangtok, Darjeeling. So obviously June 1st week is also a good option to visit.