Here's the output from the command line:
Dependencies Resolved

===================================================================================================================================================================================
Package Arch Version Repository Size
===================================================================================================================================================================================
Installing:
mongodb-org x86_64 3.2.10-1.el7 mongodb 5.8 k
Installing for dependencies:
mongodb-org-mongos x86_64 3.2.10-1.el7 mongodb 5.6 M
mongodb-org-server x86_64 3.2.10-1.el7 mongodb 5.5 M
mongodb-org-shell x86_64 3.2.10-1.el7 mongodb 6.7 M
mongodb-org-tools x86_64 3.2.10-1.el7 mongodb 41 M

Transaction Summary
===================================================================================================================================================================================
Install 1 Package (+4 Dependent packages)

Total size: 59 M
Total download size: 5.5 M
Installed size: 201 M
Is this ok [y/d/N]: y
Downloading packages:
mongodb-org-server-3.2.10-1.el FAILED [=====================================- ] 0.0 B/s | 3.2 MB -::- ETA
https://repo.mongodb.org/yum/redhat/7Server/mongodb-org/3.2/x86_64/RPMS/mongodb-org-server-3.2.10-1.el7.x86_64.rpm: [Errno 14] curl#63 - "Callback aborted" | 3.2 MB -::- ETA
Trying other mirror.

Error downloading packages:
mongodb-org-server-3.2.10-1.el7.x86_64: [Errno 256] No more mirrors to try.

I'm not having issues on other OSes (CentOS 7, CentOS 6, RHEL 6). Just RHEL7.

I can download the RPM with my browser just fine, so I think the issue is with a GPG key

The URL for the mongo repo is:
https://repo.mongodb.org/yum/redhat/$releasever/mongodb-org/3.2/x86_64/

On my redhat server, the $releasever variable resolves to "7Server".

Browsing to "https://repo.mongodb.org/yum/redhat/", and can see that "7" is an available option. I changed the $releasever variable to that in my mongo repo file, and then run the yum install. It asks me to accept a key, and then the install works.

Here's the request to accept the key:
Retrieving key from https://www.mongodb.org/static/pgp/server-3.2.asc
Importing GPG key 0xEA312927:
Userid : "MongoDB 3.2 Release Signing Key <packaging@mongodb.com>"
Fingerprint: 42f3 e95a 2c4f 0827 9c49 60ad d68f a50f ea31 2927
From : https://www.mongodb.org/static/pgp/server-3.2.asc
Is this ok [y/N]: y

Because I was suspicious about the key being the issue, I tried removing the mongodb-org, switching my repo back to using $relesasever, and then installing mongodb-org again. This time it worked.