Yes, but it is not trivial...
Sync iPod/iPad/iPhone with two computers
Although it isn't possible to sync an Apple device with two different libraries it is possible to sync with the same logical library from multiple computers. Each library has an internal ID and when iTunes connects to your iPod/iPad/iPhone it compares the local ID with the one the device normally syncs with. If they are the same you can go ahead and sync...
I have my library cloned to a small 1Tb USB drive which I can take between home & work. At either location I use SyncToy 2.1 to update the local copy with the external drive. Mac users should be able to find similar tools. I can open either of the local libraries or the one on the external drive and update the media content of my iPhone. The slight exception is Photos which normally connects to a specific folder on a specific machine, although that can easily be remapped to the current library if you create a "Photos" folder inside the iTunes Media folder so that syncing the iTunes folders keeps this up to date as well. I periodically sweep my library for new files & orphans with iTunes Folder Watch just in case I make changes at one location but then overwrite the library with a newer copy from the other. Again Mac users should be able to find similar tools.
As long as your media is organised within an iTunes Music or Tunes Media folder, in turn held inside the main iTunes folder that has your library files (whether or not you let iTunes keep the media folder organised) each library can access items at the same relative path from the library folder so the library can be at different drives/paths on different machines. This solution ensures I always have adequate backups of my library and I can update my devices whenever I can connect to the same build of iTunes.
When working with an iPhone earlier builds of iTunes would remove any file not physically present in the local library, even if there was an entry for it, making manual management practically redundant on the iPhone. This behaviour has been changed but it will still only permit manual management with a library that has the correct internal ID. If you don't want to sync your library between machines on a regular basis just copy the iTunes Library.itl file from the current "home" machine to any other you want to use, then clean out the library entires and import the local content you have on that box.
You should be able to transfer your purchases into a newly authorised library, backup the device, recover any non-store purchased media with the tips in this post from forum regular Zevoneer, (skip this if you've already got all the media) then restore the device from the backup you just made. That should preserve all your data and switch the assocation of the device to the new machine. I've had positive feedback on this method from other users but I've never had to test it personally.
First of all I think it is stupid that you can only sync to one PC, I for one have my desktop, as well as my surface pro tablet and an iPad air, and it would be nice if I was able to manage my stuff my way on my devices, I understand why they might wish to restrict the number of PCs that you can sync your iOS devices too, but one device is simply far too limited for today. Specially if you do suffer a hardware or software problem, my desktop is currently acting up with a defective video card and I am using my surface as my primary PC device at the moment. This leaves me without access to my main library and as a result I found I could not add a single song that I wanted off my surface to my phone without erasing all of the stuff on my phone, which is just stupid when access to the main library is limited at the moment.
All that said, and my hate for iTunes and its stupid rules and restrictions aside, I must thank you turingtest2 for using your advice (repeated several times in this thread I noticed) I was able to access my desktop long enough to get a copy of the .ltd files onto a flash drive, and have managed to get my song onto my iPhone without erasing all the data.
The biggest problem is that people (like myself) are probably unaware of the limit to syncing, I know I did not expect iTunes to stop me adding a single song to my phone, and as a result never thought to save a copy of the library files needed until now. And as a result if my PC had suffered a terminal problem and I was unable to access those files where then would I be? I would not have access to the main library, and I would have no way to add additional content to my device. Again it just seems like a stupid and archaic thing that needs to be revisited in this time of multiple computing devices.
Thanks again turingtest2, signed up just to show my thanks
FWIW I agree, it ought to be much easier. For those who do find themselves in the inconvenient position of trying to sync with a new computer I have a user tip that replaces the last link in my post above. Recover your iTunes library from your iPod or iOS device.