You can wait for the fade in to finish first by passing in a callback like this:
$('element').fadeIn(function () {
// Fade in complete
});
When the fade in is complete, you can continue with the delay and fade out.
$('button.other-block').click(function () {
$(this)
.find('.specific-block')
.fadeIn(function () {
$(this).delay(2500).fadeOut(1600, complete);
});
});
If it doesn't work then switch over to Mootools ;)
EDIT: working example http://jsfiddle.net/tbleckert/RcL5y/