I am having the exact same issue, I am using search head pooling and everything else seems to be working properly, when I ping the NFS where the job dispatch directory is, I see approx 1 ms of latency, the RAID1 drives are rated together to give me over 12000 IOPS
6.0.3, utilizing search head pooling
[JobManager module] Splunkd daemon is not responding: ('Error connecting to /services/search/jobs: The read operation timed out',)
I have seen this happening when there is a high count of jobs in dispatch. If you are able to, run clean dispatch command to delete jobs older than X days or X hours (choose a number for the days or hours)
If you're in a pinch, delete everything in the dispatch directory but be aware that you may have some artifacts used by dashboards or saved by users that will be deleted. The search can always run again but that might not be ideal.