All hard-disks have a 100% failure rate, it is only a matter of when.

If the problem occurred between the memory-card and the hard-disk - during the original copy from the camera, or via a card-reader, in other words - then it should have been caught by checks at that time. It is possible (though unclear from the OP) that the backups were all valid copies of what was on the hard-disk, but that what was on the hard-disk is not what was on the memory-cards.

The first rule for a backup is to immediately check it's validity and that it can be restored. That was unfortunately not done in this case, so the copied files were apparently simply six copies of an unusable file. It should be possible to choose an option in your backup software to compare the backup to the original, at the time of making the backup. If this isn't possible then change the software. Note that changing the software will involve remaking all previous backups so that they can be read under the new system.

There was no mention of offsite backup in the OP's post. If the location is destroyed then so are the backups. Ask the guys running the Fukushima reserve power-supply system how that works out.

The OP doesn't mention which O/S and backup software he is using, or which file-system. If using simple file-copies to make "backups" then the result absolutely must be checked manually (possibly using a batch or script, depending on your O/S) every time, as there is no automatic check with simply copying a file.

It is perfectly possible and reasonable to run a live backup permanently, either via disk-mirroring or with open-source tools from the Unix era. Having a schedule based on every other day, as per the OP, would only be adequate if checked at the time it was made (that should be standard for all backup plans) and when the memory-cards were over-written less often than the backup was made, so that the short-term protection for a faulty backup is to re-copy from the card and compare the files.

As the size and name are visible, you could try different programs for opening the files, file-recovery software, a different operating system etc etc. There is a chance they may be read, depending on the file system (EXT3 would be good), but not a large one.

A further necessary precaution, if no specific cause can be found, is to replace the hard-disk where the problem was originally seen and reinstall the operating system with all current service-packs and patches. The questionable disk can be checked with the SMART tools built in to the firmware and maybe, if it comes up as perfect, later re-used.

EDIT: Sorry about the blah blah. This is my day job.