I think this is because this numerical integral can not be done at machine precision. Try making your constants exact (easy to do in this case) or give them higher precision, for example like this:
c = 3*10^8
H = 7*10^5
\[Tau] = 47*10^-6
Fd = 7200
Dd = 15000
Tp = 50*10^-6
a = 52*10^-9
b = 15*10^-8
m = 256
\[CapitalDelta]f = 32*10^5
v = 7500
t = 1*10^-6
And then run the numerical integration at a higher working precision (for example 20 digits):
With[{k=5},NIntegrate[Exp[-a*x^2-b*y^2-Pi*(\[CapitalDelta]f^2*(t-k*Tp-x^2/(c*H)-y^2/(c*H))^2+2*Dd*(Fd*v*(t-k*Tp)/H)*(1-x^2/(2*H^2)-y^2/(2*H^2))+\[Tau]^2*(Fd*v*(t-k*Tp)/H)^2*(1-x^2/(2*H^2)-y^2/(2*H^2))^2)], {x, -Infinity, Infinity }, {y, -Infinity, Infinity },WorkingPrecision->20]]
During evaluation of In[61]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
Out[61]= 4.1208172836834651012*10^-865439
You will have to experiment with the working precision to get a result that works in this case.
