N-mos will not go into cutoff
The threshold voltage is specified as a certain gate voltage for a certain drain current. The current used varies from datasheet to datasheet. For example, for the PMZB200UNE, the following is in the datasheet:



The datasheet says that at somewhere between 0.45V and 0.95V the drain current will reach 0.25mA (assuming room temperature). You can safely assume the drain current will be less than 0.25ma at (say) 0.4V. You are interested in the sub-threshold behavior of the MOSFET. This particular one happens to have curves describing the behavior: 

 

From this you can see that if you need the drain current to be fairly negligible (~1uA - which is comparable to the maximum guaranteed leakage) you should limit the gate voltage to less than about 200mV. 

This is a particularly fully-specified part, with maximum Rds(on) guaranteed at 5 different Vgs values- as low as 1.5V at 25°C. 

If you are seeing relatively high current with a MOSFET with higher threshold voltage at only 30mV, then maybe your voltage source has some noise on it.