Hello, my name is Dan.
The short answer and one you may have looked at already is to check the setup for the e-mail accounts and compare them between yours and theirs.
That is where I would suspect and the first place to look.
The next thing I would recommend is repairing the permissions, this is done with Disk Utility found in Applications/Utilities, it looks like this.
With four separate accounts things can get out of whack.
The reason for this is below.
Think of your computer as a set of file cabinets in an office with a private file cabinet for each employee, call them Chris and Sue.
Each employee's cabinet is locked, unless they grant specific access to drawers inside the cabinet. Each persons private cabinet has a public outbox where anyone can see what is inside but can not put any thing in, a public inbox where anyone can put files, and an open 'shared' place where files that can be used by all.
There are three categories of users for each cabinet:
1- The owner.
2- Group members and guests who have access to one or more drawers inside the cabinet.
3- Others who have access to the in and out boxes but no access to the owners drawers.
You use these categories to define Read and Write, Read only, or no access.
Repairing permissions makes sure that the correct people and applications have the correct access to the cabinets.
If these get changed (software updates do do this) or corrupted then things go where they are not supposed to and it can mean applications won't launch or run or not preform correctly.
This is especially true with system updates because if the update info goes to the wrong place it can lead to a total system crash.
This holds true even if you are the only user.
Try that and let me know please.