This is how Bitcoin was intended to be used from the beginning, and how it's always been implemented in the reference client, but unfortunately due to a usability bug the most popular wallets these days reuse addresses and are teaching new users this incredibly destructive habit.
The problem started in the original client where it generated new private keys randomly. This meant effectively there was no way to implement a sane backup policy. Later on, they implemented the ability to generate private keys in batches of 100. This is better, but still problematic because your backup have to be refreshed after no more than 100 operations.
Obviously that was a usability disaster, and so wallet designers worked around that by just using the same private key over and over again. Other people worked out secure deterministic address generation so that you could just backup a single seed once, but by the time that got standardized (BIP32) the damage was already done.
Blockchain.info is the most popular wallet, and they've taught about a million newbies to treat single-use payment codes as permanent account numbers. It's a privacy and security catastrophe, but those issues are invisible to most people who only care about ease of use.