What is meant by "highly correlated test statistics"? as in "The motivation for developing this resampling-based procedure was an actual problem in meteorology, in which almost 2000 hypotheses are tested simultaneously using highly correlated test statistics".
http://www.sciencedirect.com/science/article/pii/S0378375899000415
Can you give an example?
 A: I don't think they are talking about dependence inherited from comparisons against a common control.
With multivariate data in meteorology, I am guessing that the variables are highly dependent, even before constructing any test statistics. Functional data are this way, with adjacent observations having near perfect correlations.
If you construct test statistics (eg, two-sample t), one for each variable, the test statistics inherit the correlation structure of the raw data, leading to highly correlated test statistics. This is probably the kind of dependence they are referring to. 
Comparisons against a common control are dependent, but this dependence is rather minor as regards its effect on multiple comparisons. Bonferroni is nearly fine here. 
On the other hand, dependencies induced by extremely highly correlated functional multivariate data do have dramatic effects, making Bonferroni very conservative.
By resampling the data vectors appropriately, you can incorporate such dependence structures accurately, and in some cases consistently. 
Bonferroni is necessarily inconsistent, with true FWER levels converging (as n -> infinity) to values less than the nominal alpha (eg, .05). On the other hand, with appropriate resampling, the methods have true FWER levels converging to alpha; this is what is meant by "consistency."  
This is all spelled out in detail in my book, "Resampling-based Multiple Testing: Examples and methods for p-value adjustment." (Wiley-Interscience, 1993)
See also 
"Pointwise Testing with Functional Data Using the Westfall-Young Randomization Method" Dennis D. Cox and Jong Soo Lee , Biometrika , Vol. 95, No. 3 (Sep., 2008), pp. 621-634 
and
Ann. Statist., Volume 39, Number 6 (2011), 3369-3391. "Asymptotic optimality of the Westfall–Young permutation procedure for multiple testing under dependence", Nicolai Meinshausen, Marloes H. Maathuis, and Peter Bühlmann
