It depends what backup program you are using and in particular what method - (full, incremental, differential, etc)
If it's a full backup every time, you should be able to delete any of the older backups you wish.
If it's differential (a set starts with a full backup, then each subsequent backup records all the changes since that full backup) you can delete any of the differential files, just don't delete the full backup which started the set.
If it's incremental (the set starts with a full backup and then each subsequent backup only records the changes since the previous incremental backup) you can't delete even one of them or you break the chain.
"As some of these backups were a few months old I thought I would do a bit of tidying, so I deleted some of the older sets."
If you delete a complete set, it should not disturb any other sets. When you deleted them maybe you accidentally deleted part of the current set, in which case the next daily incremental should show an error message because one or more of the current set is missing.
Or another possibility: Maybe it went wrong because you deleted it using Explorer and the backup program prefers you to delete sets using its own interface, so it can keep track of what's going on. If so, it's not a very robust program because it should be able to adjust to that.
Anyway, maybe it's just as well you began a new set starting with a full backup, which you should do occasionally. Otherwise incremental sets can grow very long, with many files, every one of which is necessary if you ever need to do a complete restore.
"I have never tried to restore anything!"
That's a worry.
Like having a fire drill occasionally to test the alarms and procedures, you should do a test restore of a new backup system at least once after you start using it. For a test restore, you don't restore to the main data disk which you have been backing up, in case the restore doesn't work properly and you scramble the original files. So you restore to a spare disk and then compare that to the main disk.