The "specs" are likely for no heat load, the graph below is likely the information for the same module but there is no way to be sure. Either way, the graph shows the relation between heat flow and temperature across a module (if not the module, if you want to be sure, go find the manufacturer and see if the data is supplied). As the heat load increases, the modules ability to drive the cold side to lower temperatures drops.
The coldest temperature can be determined by the heat load and the ambient temperature.
For example, with 5W of heat pumped and an the hot side of the peliter temperature of 30C, the deltaT of the module would be 80C and the coldest temperature would be -50C.
With 10W of heat pumped, the hot side of the peliter at 30C yields a deltaT of 60C. The lowest temp would be -30C
With a 15W load, the hot side of the peliter at 30C, deltaT would be 40C, the lowest temperature would be -10C
These numbers above are assuming that you have a fan that can make the hot side of the peltier the same temperature as the air, which won't happen. The fan and heatsink will always be a little bit higher than the air temp, which will further reduce the peltiers ability to get cold, because the hot side will be 5C, 10C or even higher than ambient.
I think -20C sounds about right for an ambient temp of 30C. If your ambient temp is more than 30C, then -20C would be great.
You might have a heat load in the 5W to 10W range and the hot side of the peliter is hotter than you think it is (measure it). Another thing is modules are fragile and easy to damage, it's easy to get cracks in the junctions and then they either don't work at all or have less heat pumping capacity