Q: Is there a limit on downloading previously purchased songs?

No, if you are in a country where re-downloading music is possible (it's not available in all countries), then as long as the tracks remain in the store (content providers occasionally remove items) then you can re-download them as many times as you like.

But you should also keep and maintain your own backup of all your purchases, just in case they are removed from the store.

Has a different iTunes account been used on that Mac to download past purchases, automatic downloads or iTunes match ?

90 day association : http://support.apple.com/kb/HT4627

I think what the message is saying is that if you download past purchases for that account then you will be tying the Mac to that account for 90 days - I was thinking that it was saying that it's already associated with an account, not that you are about to associate it. If you don't want that to happen (which if it's your only account then it shouldn't be an issue), then you should be able to copy the content from your other Mac to it via, for example, home sharing : copying content between computers.

Here is an important point to remember for all of us who have lost hundreds of dollars worth of iTunes music (not to mention iTunes handy little "auto-delete feature"): You can play CDs thousands of times without ever having to buy it again, and nobody will maintain and sell a list of what you listen to.

Just saying...

all of us who have lost hundreds of dollars worth of iTunes music

You mean because you neglected to keep a backup as Apple recommends?

"... back up your iTunes library. " (https://support.apple.com/HT201272). http://www.apple.com/legal/internet-services/itunes/us/terms.html - "Some iTunes Eligible Content that you previously acquired may not be available for subsequent download at any given time, and Apple shall have no liability to you in such event. As you may not be able to subsequently download certain previously-acquired iTunes Eligible Content, once you download an item of iTunes Eligible Content, it is your responsibility not to lose, destroy, or damage it, and you may want to back it up."

Back up to where? I my years of experience local iTunes backups are often not recognized (i.e. backup the backup to another drive in case the main drive crashes), as they are apparently presumed to be the collection of somebody else.  Back up to Apple?  Oddly, having bought the music I don't feel like it's theirs--why should I let Apple track and sell my personal activities, what songs I listen to and when?  I is really none of their business.

Maybe you and I should just let them install cameras and microphones in every room in my house so that they can really get to know me?  Maybe I will carry something on my body that tracks and reports my physical movement 24 hours a day.  Hey, it could be a watch...

Oh.

I could backup to removable disks--oh wait, I do, they're called "Compact Disks" (or "CDs") and I don't have to have Apple's permission to listen to the music.  I don't have restrictions on the number of times I can listen to them.  I don't mysteriously lose songs as I do in the iTunes library--whether bought from the iTunes strore or copied from my CDs.  No, the only real way to make sure you have to buy a song only once is the good 'ol CD; everything else is surrendering your personal life to apple.

TigerRider wrote:

 

Maybe you and I should just let them install cameras and microphones in every room in my house so that they can really get to know me?  Maybe I will carry something on my body that tracks and reports my physical movement 24 hours a day.  Hey, it could be a watch...

If you think that would help Apple to help you not lose your music tell them: apple.com/feedback