The best we ever got was:
if (webEngine.getLoadWorker().getException() != null && newState == State.FAILED) {
exceptionMessage = ", " + webEngine.getLoadWorker().getException().toString();
}
but that didn't help.
(Our error was caused by a missing CookieStore, it seems you don't get one for free - and have to set a default one: http://docs.oracle.com/javase/7/docs/api/java/net/CookieHandler.html)