This is strange; the code above works perfectly on my machines (Win7 / MMA9) and (Linux / MMA10):

In[1]:= m = {{1, 2, -3}, {0, -1, -6}}
m[[2]][[1]]
m[[2]][[1]] = 3
m

Out[1]= {{1, 2, -3}, {0, -1, -6}}

Out[2]= 0

Out[3]= 3

Out[4]= {{1, 2, -3}, {3, -1, -6}}

@Daniel Lichtblau: Thank you for pointing out the principal difference in the notation [[n,m]] and [[n]][[m]]! Once one knows it, it becomes quite obvious ...

You need to use the unnested form e.g. m[[2,1]]=3. As an rvalue (in the CS sense) m[[2]][[1]] is fine, but it does not work as an lvalue (that is, something to which one can assign a value) in Mathematica.