TL;DR: If you have a frequent recurring backup you are safe, otherwise use a deterministic wallet.
No, you are not safe in that scenario.
Let me rephrase that, you are not safe if you really want to use your the coins from your addresses.
Wallet.dat file stores all public and private keys of your current addresses but standard client generates a new address every time you generate some change spending your money and those addresses will not be saved in your distributed copies.
Also if you generate a new address manually it will not be in your backups.
So if you recover the file from a backup those "change" addresses will be lost and you will not be able to use that coins.
Well that is not really true :D indeed to leverage this kind of problems with recent backups standard client (and probably doge client) pre-generates a bunch of addresses (100) in a qeue and when it needs a new one (used for change or when you ask for it) it just take the first one and add one to the end of the qeue.
So while it is not really safe to store a backup forever if you backup frequently you will have no problem recovering all your coins when you recover a backup.
Another option is using a deterministic wallet, it is a system that given a unique passphrase, file, code or whatever will generate always the same addresses with their private keys, so as long as you save or memorize the seed you can recover everything.
bitcoin-qt (and dogecoin-qt) does not support this but clients like Armory and other online wallets do.