Unless I am missing some famous Linkedin engineering blog post explaining it, I can make an educated guess about them using S3.
What are the dynamics of them not using Azure but AWS? Existing architecture pre-dating the MS acquisition?
When S3 goes down, sites may have their traffic fall over to sites with higher latency or lower band width or both or just another cloud. That will create congestion and congestion may cause an activity such as transferring a large file time out. And it may happen at any level, for example a service using your ISP has a fall over plan from S3 to their local servers and this spikes your ISP's traffic and your Linkedin session has a lower service level agreement. That same situation can occur at every point in the network and has nothing to do with Linkedin.