Are periodic full backups really necessary on an incremental backup setup?

I intend to use an old computer I have as a remote backup server for myself and a few other people. We are all geographically separated, and the plan is to do incremental daily backups using rsync and ssh.

My original idea was to make one initial full backup then never again have to deal with the overhead of doing it, and from that moment on only copy the files changed since the last backup.

I've been told that this could be bad, but I fail to understand why. Since each snapshot is comprised of hard links to the unchanged files plus the original changed ones, isn't it going to be identical to a new full backup? Why would I want to make another full backup?

EDIT:

I should have explained a point better. When I mean I'm going to do incremental backups using rsync, I mean this:

rsync -avh --delete --link-dest=./remote/previous_increment ./local/ ./remote/new_increment

Which results on a complete and working snapshot, since it will contain hardlinks to all the unchanged files. Even if the full backup and all of the previous incremental ones happen to be deleted the last incremental is still consistent.But I'm pretty sure that if any of the previous files get corrupted, the last ones will too, since they point to the same inode.

What if I do a synthetic full backup server-side periodically, by breaking the links on the last snapshot and copying it to another, write-protected HD (say, once a month). That way I would have a redundant full copy and would still avoid the overhead of re-sending the files.

Would that solve the issue? Would I still need to do full backups?