Is this file on a shared network volume, or hosted with FMS? I'd guess it's on a shared network volume because your issues are EXACTLY those that FMS solves.
FMS can backup files while they are opened by multiple users, and can keep multiple backups. All automatically. No scripts needed to close and reopen files or worry about idle time.
Sharing FMP files on network volumes is the WORST way of doing it, it inevitably leads to data corruption.
He said it was not on a server. You can do this with the Save A Copy As script step. If you're on a Mac you can use a cron job, schedule a shell script to run an Applescript that will check to make sure the file is open and start the backup script from the file itself. The FM script you can name your file variable with the date and time then they will always be different.
I did things like this before. Much easier to fight with IT and get a FileMaker Server.
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I agree with Agnes. Server was built for this. The impact on your hosting will be much less and it will be more reliable, stable and cope much better with traffic. The logs are better the backups are built in... there are so many reasons.
If you absolutely have to stick with FMP. I suggest your database itself can schedule the backups. You don't have to resort to externals just make each script have a check for the time and save a copy if > the last backup time (which you store) + the interval you want. If there are no scripts running, there is nothing much new to back up...
Take a look at this article from FileMaker Hacks:
This is the method i use to create backups of files i am working on before they get uploaded to server. Normally i setup a script with InstallOnTimer to run every hour while im working on the file, but im sure it can be scripted differently for your needs.
I am going to make an assumption that this is relatively unimportant data since you are using peer-to-peer sharing from client app. Matt Leach has a good link for you.
If you ever decide your data is important enough to need real server services, FileMaker Server is designed to handle these types of things and much more. Not only can you get regularly scheduled backups, you can get progressive backups. FileMaker server is faster and designed for multi-connections and you don't have to leave a copy open on some client machine. Additionally, you can encrypt all data between clients and the server, which is not possible with a peer-to-peer system like you have described. Not to mention that you can run scheduled scripts from the server.
If things are not too tight financially, I would seriously take a look at FileMaker server since it is designed to meet these needs. It'll make life much better for you. But if you can't, I understand and that is how I first ran FileMaker databases.