It's probably outside what NSolve methods can handle. Might try using FindRoot or FindMinimum of a sum of squares to get isolated real roots.

I am not able to replicate the problem, at least in Mathematica 10.0.2.

exp1 = 
  ComplexExpand[
    Im[(1 + I)/(2 + 3 I - bb) + (1 + I)/(2 + 3 I - cc)] + 
     Im[(1 + I)/(2 + 3 I - dd)]] - 3.5;
exp2 = ComplexExpand[
    Im[(2 + 2 I)/(3 + 4 I - bb) + (2 + 2 I)/(3 + 4 I - cc)] + 
     Im[(2 + 2 I)/(3 + 4 I - dd)]] - 0.75;
exp3 = ComplexExpand[
    Im[(3 + 3 I)/(5 + 7 I - bb) + (3 + 3 I)/(5 + 7 I - cc)] + 
     Im[(3 + 3 I)/(5 + 7 I - dd)]] - 0.75;
eq5s = {exp1, exp2, exp3};
varr2 = {bb, cc, dd};
Timing[solns = NSolve[eq5s, varr2];]

(* Out[20]= {2.727674, Null} *)

In[21]:= Length[solns]

(* Out[21]= 48 *)

Earlier versions used a different mathod and may be slower. One thing that works in version 9 is to clear denominators.

polys = Numerator[Together[eq5s]]

Now NSolve[polys, varr2] will be quite fast.