I have calcuated this integral:

NIntegrate[Sqrt[x Exp[-100 x]] Sqrt[y Exp[-100 y]] (1 - Exp[1000 I (x + y)])/((x + y) (y - 0.001)), {x, 0, \[Infinity]}, 
{y, 0, 0.001, \[Infinity]}]

and due to the singularity at y=0.001, the integral should diverge. However, it is posible to separate the real and imaginary part, in such way that I can integrate firts wrt x symbolically, and then wrt y numerically:

Assuming[y >= 0, Integrate[(1 - Cos[1000. (x + y)]) Sqrt[ x ] Exp[-50. x], {x, 0, \[Infinity]}]]
(*0.00250663 + 0.0000182432 Cos[1000. y] + 0.0000212049 Sin[1000. y]*)

and then:

NIntegrate[1/(-0.001 + y) Sqrt[y] Exp[-50 y] (0.0025066282746310014 +  0.000018243204055544903 Cos[1000. y] 
+ 0.000021204895736310885 Sin[1000. y]), {y, 0, .001, \[Infinity]}, Method -> "DoubleExponential", 
 MaxRecursion -> 200]
(*0.000567425*)

Similarly for the imaginary part, I obtained 2.21313*10^-6 I.

Threfore, my integral value is 0.000567425+2.21313*10^6 I.

My question is, if my integral diverges, how much reliable my numerical result is as I did not get any error from MMA? Oh, I am using MMA ver 11.0.1 in a iMac with macOS Sierra ver. 10.12.1