  In the analysis of microarray data, and in some other contemporary
statistical problems, it is not uncommon to apply hypothesis tests in a highly
simultaneous way. The number, $\nu$ say, of tests used can be much larger than
the sample sizes, $n$, to which the tests are applied, yet we wish to calibrate
the tests so that the overall level of the simultaneous test is accurate. Often
the sampling distribution is quite different for each test, so there may not be
an opportunity for combining data across samples. In this setting, how large
can $\nu$ be, as a function of $n$, before level accuracy becomes poor? In the
present paper we answer this question in cases where the statistic under test
is of Student's $t$ type. We show that if either Normal or Student's $t$
distribution is used for calibration then the level of the simultaneous test is
accurate provided $\log\nu$ increases at a strictly slower rate than $n^{1/3}$
as $n$ diverges. On the other hand, if bootstrap methods are used for
calibration then we may choose $\log\nu$ almost as large as $n\half$ and still
achieve asymptotic level accuracy.
  The implications of these results are explored both theoretically and
numerically.
