If you end up having to "roll your own", this article might be helpful:

"A Note on Computing Extreme Tail Probabilities of the Noncentral T Distribution with Large Noncentrality Parameter" by Viktor Witkovsky, Acta Universitatis Palackianae Olomucensis Facultas Rerum Naturalium Mathematica (52) 2013 131-143.

Good, thanks. I will practice pasting code next time. I think this is a bug. I just sent it to Wolfram support and I attached that notebook (btw, I see that one could attach files here too). I noticed the underscores, but I thought I had fixed them by hand.

I am not sure this is pushing the function too far. It is odd, but sometimes I get better results for larger numbers (Mathematica seems to change its mind about the method to use, I recall seeing an example to this effect somewhere in the 'Help'). I have plotted the time it takes to make similar calculations (as a function of n, say) and it does very strange things, not random, but difficult to explain.

I am pretty sure Mathematica can do better. I often work with the regression model functions (LinearModelFit and NonlinearModelFit). These functions can provide tables with statistical summaries of the regression models (p-values for F- and t-statistics, etc.). It is quite challenging (but possible) to reproduce the results in those tables using the CDF and InverseCDF functions, but those Fit functions evaluate quickly and print the tables in a heartbeat. It seems clear that there are statistical packages with more efficient versions of the CDF functions somewhere in the code.

Anyway, thanks for your attention and feedback. Wolfram sent me an automated reply saying they will get back to me within 3 business days. I will update this post if they provide any useful work-around.

OL.

To make a specific example, does the following capture the issue? (This is with Mathematica v. 10.1)

(* Set values *)
\[Nu] = 4200;
\[Alpha] = 0.95;

(* Get t-value *)
t = N[InverseCDF[NoncentralStudentTDistribution[\[Nu], 0], \[Alpha]], 50]

(* Plug value back into CDF function *)
N[CDF[NoncentralStudentTDistribution[\[Nu], 0], t], 50]

with output

1.64522

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in
Statistics`NoncentralDistributionsDump`z$163348 near {Statistics`NoncentralDistributionsDump`z$163348} = {64.8939}.
NIntegrate obtained 1.746085001809244`15.954589770191005*^6694 and 6.297643634774244`15.954589770191005*^6690
for the integral and error estimates. >>

0.95