Simple question - will I ever be able to store my bitcoin with an organisation who will provide secure storage for it, whilst at the same time allow me easy access to it (like a bank currently does with my cash)?
Not a simple answer. Your cash is safe because the FDIC (Federal Deposit Insurance Corporation -->
www.fdic.gov) insures your usd in a FDIC bank (which all of them are). We don't have, and possibly don't want btc being insured by the FDIC.
With security you get less liquidity. For every btc taking part in some networked system in which it can be transferred on a moments notice, is going to be under risk from threat agents. When you move btc away from networked systems and into a storage situation there is substantially less, but different risk, and at the same time a liquidity problem rises up.
I wouldn't trust anyone with holding an amount of btc on my behalf that I couldn't walk away from with little regret.
Here is a little tip - You must take a risk based approach to how you hold your btc (this is from an older post of mine):
1 - Quick access spend thrift account (no more than 200 usd exposure) - Windows XP/7 with standard bitcoin software. Password protect your login and make sure your running some decent anti-virus
2 - Medium access large account (no more than 5,000 usd of exposure) - VMware image of Ubuntu JeOS with OS based full disk encryption. Has no services running (if it's running apache you fail) and a stable release of bitcoind. Connect this only to known trusted nodes when needing to move coins
3 - Hard access mother account (all those secret coins generated the first year) - Every possible disk that ever held a private key is suspect. Almost all of those should have been wiped with random data 5 times. <-- before doing that, account key pairs should be replicated to 3 types of medium (dvdrw, flash drive, usb ext. hd) and each of these medium would be placed at separate bank security deposit boxes