Quick answer
No, but it rains far more on the north coast (São Vicente, Seixal, Santana average 150+ rainy days a year) than the south (Funchal averages 88). Most days have rain somewhere on the island while elsewhere is sunny. Drive 20 minutes through the mountains and the weather usually changes completely.
Context
Madeira has a sharp microclimate split. The mountainous spine catches the north-east trade winds and pushes moisture up the northern slopes, where it falls as orographic rain. By the time the air descends to the south coast it's warm and dry, Funchal sees about 88 rainy days a year, less than London.
Practical rule for a holiday: if it's wet where you are, drive to the other coast. Cloud often sits on the high peaks (Pico do Arieiro, Encumeada) all day while Funchal stays bright. Conversely, when the south is cloudy, the north can be sunny.
The wettest months are October to February. The driest are July to September. Even in winter, two-thirds of days in Funchal record under 1 mm of rainfall.