...if a new block is added to the chain approximately every 10 minutes
10-minute block times are for the Bitcoin blockchain. In Ethereum the block time is (currently) about 18 seconds. Other blockchains will have different block times depending on their design.
For example, if I send money to someone, might it take up to 10 minutes for them to receive it?
(Substitute "10 minutes" for "18 seconds" to bring the conversation back to Ethereum.)
It will take at least that amount of time (on average*), and maybe even longer. There is no way to send money instantaneously.
(*Block times are average figures. There will be a bell curve around this value.)
When you make a transaction, it gets put into a transaction pool waiting to be incorporated into the blockchain. Your money is only "sent" once the transaction is incorporated into the chain. If there are lots of transactions, the transaction pool will be full to bursting. Only so many of those transactions will fit into a block. The miners decide which transactions to take from the pool and put into the block they're mining.
(In Ethereum the number of transactions that will fit into a block is dictated by the block gas limit, so they can't just put all the available transactions into a block to make a huge uber-block.)
Pre-reading: What is meant by the term "gas"?
Each transaction will have an associated amount of gas, which is what the person sending the transaction is willing to pay for the transaction to be included in a block. (That's a very basic description, but enough for this explanation.)
At times of normal use, your transaction may well be incorporated into a block in 18 seconds. However, when the number of transaction in the pool exceeds what can be put into a single block, you will inevitably have transactions that have to wait around. If they're waiting around, then the miners will start being selective, and will pick transactions that have a higher associated price. The miners want to make money: you've entered into a market.
So in short, if there's competition in the network, the amount of time you have to wait for your money to be sent will be inversely proportional to the fees you're willing to pay.