R allows what are called generalized linear mixed effects models. In these, the response variable is allowed to be from a few different families, including binomial (which, if coded as 0 and 1, gives logistic regression). 

The function used to be called glmer(). I'm pretty sure that now more recent versions of the regular mixed effects models function lmer() allows you to specify a family (e.g. 'binomial') and a link function (e.g. 'logit'). lmer() allows the specification of random effects and nesting. You can find more info on Doug Bates' slides, in particular the very last one, <a href="http://www.unil.ch/webdav/site/ee/shared/WorkshopD.pdf"> here </a>. He wrote lmer(), so I believe him when he says it works.

Keep in mind that you need numerous (more than 6 or so) different 'subjects' to be able to estimate random effects efficiently.