Just a data point...
July 5th: Had a damaged bag and had to make a complaint via email (nobody answered the Swedish phone number listed on the website) on July 5th.
July 6th: Received a personal response asking for further information
July 16th: Canned response saying that it would take awhile to get back to me
August 5th: Response on the complaint and voucher for a new bag


1 month isn't great, but Customer Care did respond the next day to my initial complaint. Looks like vacations started just after that, but I was surprised to get an email in between warning me of the delay.

My biggest complaint would be with the staff working the phones. SAS lists a number to call, but nobody answers. I called the Gold line and was rudely (and incorrectly) told that I had to declare the damaged bag at the airport (O'hare) and that I was not entitled to compensation. It might have been the first time that I lost my cool and the agent ended up hanging up on me. They need to train these agents better as what I was being told was clearly wrong and definitely not in agreement with what is listed on the website. Sadly the agent was not willing to look at the rules listed on the website, decided to agitate me instead of help and decided to hang up on me (she even told me she was going to do it) rather than hand me over to someone else (as I had asked). Still waiting for a response to my separate complaint on her behavior, but I suspect that will take much longer.