As you define it

In[1]:= sumnint[T_] := 
 Sum[cn[n] NIntegrate[
     Exp[-an[n] x T] /. {an[1] -> 1, an[2] -> 2}, {x, 
      0, \[Infinity]}], {n, 1, 2}] /. {cn[1] -> 1, cn[2] -> 2}


In[6]:= sumnint[1]
Out[6]= 2.

In[10]:= sumnint[q] /. q -> 1
During evaluation of In[10]:= NIntegrate::inumr: The integrand E^(-q x) has evaluated to non-numerical values for all sampling points in the region with boundaries {{\[Infinity],0.}}. >>
During evaluation of In[10]:= General::stop: Further output of NIntegrate::inumr will be suppressed during this calculation. >>
Out[10]= 3 NIntegrate[Exp[-an[n] x] /. {an[1] -> 1, an[2] -> 2}, {x, 0, \[Infinity]}]

Mathematica stops working after the NIntegrate::inumr message. That's all. One way to circumvent that is to define

  sumnint[T_?NumericQ] := <snip>

which makes good sense if one is going to use a numeric function inside sumnint.

The other way is to keep the function undefined too, with other words

In[16]:= sumnint[T_] := (Sum[
     cn[n] X[Exp[-an[n] x T] /. {an[1] -> 1, an[2] -> 2}, {x, 
        0, \[Infinity]}], {n, 1, 2}] /. {cn[1] -> 1, cn[2] -> 2}) /; UnsameQ[T, x]

In[17]:= sumnint[1] /. X -> NIntegrate
Out[17]= 2.

In[18]:= sumnint[q] /. {X -> NIntegrate, q -> 1}
Out[18]= 2.

In[19]:= sumnint[q] /. {q -> 1, X -> NIntegrate}
Out[19]= 2.

The first icon, the one that looks like a rectangle with <> inside it, is for code samples.

It would be easier to diagnose if you had put your code into a code sample window. In the text form, it looks like the symbol for infinity is wrong. It should be [Infinity].