I was in your same situation with 14 downloads and 14 attempts.
You have to make sure that your Mac is booting into 64 bit mode. You can make sure of this by going into System Preferences and scrolling down to 'Software' and making sure that you are booding with 64 bit Kexts.
If not, restart while holding the 6 and 4 keys down until you see the spinning gear wheel and try installng again.
BUT, what I also did was to go into Applications, click on the Lion installer, Get Info, and give yourself custom priviliges to Read And Write.
I did this simultaneously, and I don't know what function did it, but it worked for me, with no further downloads of the installer.
Good luck!