The KLM and the Air France online checkin system is the same. In fact the AF checkin is hosted on KLM's servers, it's just different branding.

KLM also does the online checkin system for Czech Airlines and VLM (including before VLM became part of AF-KLM).

Of course the award for the worlds worst airline IT implementation must go to bmi for their diamond club 'upgrade'

As for making KLM checkin work, I now always do the following before I attempt checkin and it has avoided me running into random problems using Firefox - I know we shouldn't have to do this but...

1. Open up the preferences panel (Tools > Options in Windows, Firefox > Preferences in Mac, Edit > Preferences in Linux)

2. Select Privacy from the bar at the top, click "Show Cookies..."

3. Type klm into the search box at the top, now remove all the klm.com cookies in this list, including those on subdomains like www2.klm.com and secure.klm.com

This will remove all your KLM cookies without removing any other more useful cookies.