Elasticsearch throwing an “OutOfMemoryError[unable to create new native thread]” error

We just did a rolling restart of our server, but now every few hours our
cluster stops responding to API calls. Instead, when we make a call, I get
a response like this:

curl -XGET 'http://localhost:9200/_cluster/health?pretty'
"error" : "OutOfMemoryError[unable to create new native thread]",
"status" : 500

I noticed that we can still index data fine, it seems, but cannot search or
call any API functions. This seems to happen every few hours, and the most
recent time it happened, there was no logs in any of the node's log files.

Our cluster is 8 nodes over 5 servers (3 servers run 2 elasticsearch
processes, 2 run 1), running RHEL6u5. We are running Elasticsearch 1.3.4.

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