Machu Picchu is still accessiblel in January and Fenruary, it doesnt close. The Inka Trail (4day and 2 day) is closed during February due to maintanance and rain.
I wouldnt says it a definate no no. Machu Picchu is not like a beach resort where if it rains all is spoiled. The mist and rain give the ruins a mystical touch and there are much less tourist crowds during Jan-March. The weather is warmer during rainy season as well.
The best months weather-wise is spring and fall (April-May and Sep-Nov). But if you cant choose, any time is good for Cuzco and Machu Picchu.
Hope this helps and the links below have more information
Richard Leon
www.peruforless.com