Yes, you do get sunburned on a cloudy day, that is of course, if you are engaged in outdoor activities but not properly protected against ultraviolet radiation (UV).  

It is true that on a cloudy day, you won't get as much exposed to UV from direct sunlight as when it is a clear sunny day.  However, sunlight, including UV, are scattered by gases in the atmosphere, as well as by clouds, dust, haze and even fog.  This part of the sunlight that reaches the earth's surface is called diffuse sunlight.  Because atmospheric scattering is inversely proportional to the wavelength of radiation, therefore, by virtue of its short wavelength, UV is most severely scattered.  In other words, there is a significant UV component in diffuse sunlight.  

From the time series of solar radiation on 20 May 2010 (Fig. 1), which is a cloudy day with 73 % cloud amount, one can notice that although the direct solar radiation has the highest peak values when direct sunlight reaches the ground through a break in the clouds, the average intensity of diffuse solar radiation is higher.  Around noon time, the highest value of diffuse solar radiation nearly reached 600W/m2, some 70% of the peak value of direct solar radiation in a clear day (Fig. 2).  In fact, in 27 days in April 2010 (which is a cloudier month than normal), the total diffuse solar radiation is higher than the direct solar radiation (see Fig. 3).  In some cases, the diffuse solar radiation is several times more intense.

I had an unfortunate experience of getting quite nastily sunburned when I hiked in late March this year in Ma On Shan.  Although it was quite bright throughout, the fact that Ma On Shan was completely shrouded in hill fog gave me the wrong impression that I did not need to be protected against sunlight.  But the scattered UV did take its toll on me throughout my 5 hours of hiking in the fog. For real-time information of direct and diffuse solar radiation, please visit the Observatory's webpage at: https://www.hko.gov.hk/en/wxinfo/ts/display_element_solar.htm