Thanks, westie457! This has solved the problem for me - although it is really a workaround rather than a full solution.
It would appear that it is the bug in Oneiric with 802.11N (bug 836250). As you suggested, I have changed my router to use 802.11G instead of N and now all is well. Thanks again! I would have probably instead tried first the b43 drivers and then tried to disable N on the computer rather than the router. This is a much cleaner solution!
Here's a recap of the situation as I understand it to be. Perhaps this helps others with this problem or setup:
Computer setup: Dell Studio 1747 with Broadcom BCM43224 14e4:4353 wireless running Ubuntu 11.10 x64. Ubuntu suggests the proprietary STA driver for that under > System settings > Additional drivers and generally speaking that is correct. However, at this time there is a bug in Oneiric with connecting to 802.11N wireless standard. So, changing the router to use the (older and slower) 802.11G standard solves this problem. Hopefully the bug will get solved and we can switch the N back on at some point. I'll try it whenever there's an update to bcmwl-kernel-source.
Changing router configuration: usually done by going to http://192.168.1.1 In my case it's a Swisscom Centro Grande (go to > Settings > WLAN > Transmission).
Cheers.
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