the problem is usually due to 'invisible' formatting codes that are for some reason not counted until the cutting off stage. People get around this via two ways - for some people copying first to another program such as notepad and then recopying and pasting to the AMCAS form loses the formatting codes.
If this doesn't work your only solution, if you want to retain all the text, is to cut the first paragraph and retype it directly into AMCAS - if that doesn't work, continue with the second paragraph etc. At some point you will have removed all the invisible formatting. Unfortunately this sometimes ends up meaning you re-type the entire thing directly.