Not an answer but someone else that would like to know this information.
I've had very bad experience some years ago with Adobe acrobat licensing (my backup copies of the OS 'took up' licenses until the original boot drive was deactivated by Adobe - even though I never installed it on a second machine).
So I trust Adobe licensing 'promises' - not at all.
The Creative Suite end-user license reads:
the primary user of the Computer on which the software is installed under Section 2.1 may install a second copy of the software for his or her exclusive use on either a portable Computer or a Computer located at his or her home, provided that the software on the portable or home Computer is not used at the same time as the Software on the primary Computer.
Note the second installation has to be on the same platform (Windows or Mac). But if you're unable to activate a second installation, then perhaps something is somehow taking up that slot.
Your post title says "transferring to a new computer" but the text hints at having the software on both. If you want to transfer the license from one machine to the other, that's fairly straightforward. But if you want to maintain the software on both, you'll have to try to figure out what's preventing the second activation... (perhaps a third or backup installation somewhere? or something like what Rick ran into)
Thanks ProDesign, I appreciate your help.
I ended up having to call the licencing department at Adobe Customer Support.
I discovered that my Adobe licence key had been used on two computers and this had used up my quota (2 machines max). I'm not sure where that 2nd computer was (I had only installed it on one!). Nevertheless, the person I spoke to was able to deactivate the key for the two computers, so I could effectively start again.
Once she had done this the licence key immediately worked on my laptop.