How to select a text within the Highlighted text in Adobe Acrobat?

Some workaround for this very old bug.

Toggle Read Mode

On OS X: Ctrl-CMD-H

On windows: Ctrl-H

Use block selection

Shift + Alt + Drag-select

If you turn on the "Edit Text & Images" in the "Tools" panel in Adobe Acrobat, you should be able to select text inside highlighted region and right-click to do cut/copy/paste. I just tried on my laptop using Acrobat 11.0. Hope this can solve your problem. Not sure if Adobe Reader can do the same.