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Bitcoin exchanges, where you buy and sell Bitcoin, are often paired to your bank account in order to convert your money to Bitcoin as you buy it. And your bank account is obviously tied to your identity. Let's go through two different methods for taking your stores of wealth off the grid.
1. Cash deposits through LocalBitcoins.com.
LocalBitcoins is a service that connects people seeking to buy Bitcoin with those looking to sell, and vice versa.
For those looking to anonymously buy it, simply selected a trusted vendor from the list of people able to sell you the amount of Bitcoin you want. They'll provide you with banking information and you make a cash deposit into their account. Once the transaction's confirmed, you'll usually receive your Bitcoins in a matter of minutes.
Here's an example. If I want $100 in Bitcoins, I search for people able to accomodate that amount as a cash deposit. Here's what the search results look like:
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I'd probably use the first vendor, as his feedback shows over 100 completed transactions and a 100% positive rating. The description lists Wells Fargo as this person's bank, so at this point I'd go to a Wells Fargo with $100 cash and make a deposit into the vendor's account. As soon as it's confirmed, I have anonymously acquired Bitcoins in my LocalBitcoins account.
2. Use a Bitcoin tumbler.
If you already have Bitcoins that you want to anonymize, you'd probably want to use a Bitcoin tumbler. Here's a scaled-down version of the idea behind them.
If you and I each put 500 pennies in a jar and shake it up, we can then remove 500 pennies each without knowing whose were initially whose. Bitcoin tumblers facilitate groups of people anonymously pooling their Bitcoins, mixing them up, and redistributing them back to the group, usually minus a small fee for the service.
The most popular tumbler seems to be BitcoinFog, which you can read more about here.