If the documents are tagged properly, the copy/paste would work. Sounds like some of them aren't.
There is no solution using the free Reader software but you may want to try a PDF to Word converter such as http://www.pdftoword.com/
(I don't use or endorse that particular converter. Just found it doing a search and used it as an example)
Well, this may be a bit late given that I am replying more than 4 years after the question was asked... but I think I have found an answer.
My document has line numbering for aligned along the left margin for every single stupid line in this document. So after some trial and error, I used the crop tool found under Tools/Advanced Editing, and simply cropped the page so the line numbers were not included. From there, I simply copy/pasted the text I wanted into a new word document.
While I imagine this is not helpful to the original poster - unless he/she is VERY patient - I hope this answer will be helpful for future users who may encounter this issue. And while I'm at it... HELLLOOO FUTURE USERS!!! Send me a note if/when you read this and tell me what the future looks like! Are we all driving flying cars now?
The unfortunate part of this advice is that it won't work for Adobe Reader. Cropping is a function of Adobe Acrobat.
And, hello from the not-very-distant future.