# Summand is not numerical at some point

I am trying to do some numerical integration with the following codes:

EI[m_, x_] := x^m/m! (0.5772156649 + Log[x] - HarmonicNumber[m]) + Sum[If[k != m, x^k/((k - m)*k!), 0], {k, 0, Infinity}];
phi[m_, n_, x_] := (-1)^(n + 1)*m^n*Sum[(-Pi^2)^k*Binomial[n, 2*k + 1]*(EI[m, x])^(n - 2*k - 1)*(x^m/m!)^(2*k + 1), {k, 0, Floor[(n - 1)/2]}];
F[n_, m_, t_] := NIntegrate[((1 - Exp[-t*x])*Exp[-n*x]*phi[m, n, x])/x, {x, 0, Infinity}, WorkingPrecision -> 30];


They worked fine when I calculated F[10, 24, 0.4] // Timing. But when $m>24$, for example, F[10, 25, 0.4]// Timing, I got an error saying

NSum::nsnum: Summand (or its derivative) If[k!=25,x^k/((k-25) k!),0] is not numerical at point k = 15.

and forF[10, 26, 0.4]// Timing, an error saying

NSum::nsnum: Summand (or its derivative) If[k!=26,x^k/((k-26) k!),0] is not numerical at point k = 15.

Can anyone help? Thanks.

Just a extended comment: Sadly I'm not sure which version should we use to reproduce the warning mentioned in the question. (I should have recorded it 7 years ago! ) Anyway, NIntegrate gives a seemingly reasonable result in v9.0.1:
While in v12.3.1 the Throw::sysexc warning pops up. If the WorkingPrecision -> 30 is removed, NIntegrate::ncvb pops up and 3.49077*10^35 is given as output, which no longer looks reasonable.