The answer to your question really depends on exactly which updates you’re talking about – as well as exactly how they’re delivered and installed.
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Let’s start with Windows itself. The answer here is generally no.
Updates often build upon previous updates, so removing a prior update can sometimes cause problems. But there’s a a caveat: a cleanup utility – sometimes called Windows Update Cleanup – may have the option to remove prior updates.
Two things though: If you’re going to do this make sure to use Windows disk cleanup; it will keep track of those dependencies that I mentioned, where a new update relies on an older update.
And if you do this, keep in mind that you will be losing the ability to uninstall any updates that remain. Currently, you can uninstall an update, which basically means that Windows replaces the current updated files with the old ones from the previous version. If you remove those previous versions with a cleanup, then it just can’t put them back to perform the uninstall.
This same scenario is also generally true for Microsoft Office: updates are applied to the program directly. Files are either removed for you at that time or if they’re not, they’re probably needed to keep things working.
Other programs, however, often update themselves quite differently. Rather than just updating parts of the program or patching the program, they simply have you download a completely new version and install it.
When this technique is used, you can typically delete the previously downloaded installer.
Note that you’re not deleting the installed program that’s been updated by the new download. What you can delete is the old download from the original installation, or from the previous update, and this applies only when the program updates itself by downloading an entirely new copy.
And for the record, I do recommend that you save at least the most recent download installers, especially for programs that you may have purchased. Should you ever need to reinstall from scratch, you’ll have them ready to go.
Now, if it’s not clear that a program updates in this manner – a complete download and replacement – then honestly, I’d do nothing. I’d leave well enough alone. Chances are the program is either cleaning up after itself, or the parts that remain are required for some reason.