Looks like you have networking issues. I know that's rather stating the obvious though. If the cluster nodes cannot communicate then the partitions will race for thew quorum device (or server). This node lost. When the node reboots, if it cannot communicate with the other partition it will be prevented from joining the cluster.
I'm not sure quite where to start. First, it would be useful to know what hardware you are running on. If it isn't a supported configuration, then you might have run into something that is hardware specific. If it is a supported configuration, then I would check that the switches are properly configured. In particular, are the nodes connected to two separate switches and do they use isolated (private) vlans for the private networks. If there is some transient event occurring on the switch, like a broadcast storm that takes >10 to recover from, then the cluster nodes will be affected. If the switches use spanning tree and they are trying to reconverge after some configuration (topology?) change, then again, unless that happens in < 10 seconds, the cluster nodes could be affected.
Sorry, these are rather vague suggestions, but it's difficult to guess what might be the root cause.