Please add the troubleshooting information to your post To find the Troubleshooting information:

• Open Help (or click on three-line-icon and select Help)
• Choose Troubleshooting Information
• Use the button Copy to clipboard to select all. Do not check box "Include account names"!
• Paste this in your post.

Application Basics

   Name: Thunderbird
   Version: 45.2.0
   User Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.11; rv:45.0) Gecko/20100101 Thunderbird/45.2.0
   Profile Folder: Show in Finder
   Application Build ID: 20160630070546
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory

 Mail and News Accounts
   account1:
     INCOMING: account1, , (pop3) incoming.verizon.net:995, SSL, passwordCleartext
     OUTGOING: , smtp.aol.com:465, SSL, passwordCleartext, true

   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

   account5:
     INCOMING: account5, , (imap) imap.aol.com:993, SSL, passwordCleartext
     OUTGOING: , smtp.aol.com:465, SSL, passwordCleartext, true

 Crash Reports
   http://crash-stats.mozilla.com/report/index/bp-8858655b-f4b5-42ee-85d1-ad8da2120602 (6/2/12)

 Extensions
   Lightning, 4.7, true, {e2fda1a4-762b-4020-b5ad-a41df1933103}

 Important Modified Preferences

   Name: Value

     browser.cache.disk.capacity: 358400
     browser.cache.disk.filesystem_reported: 1
     browser.cache.disk.smart_size_cached_value: 358400
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     dom.apps.reset-permissions: true
     extensions.lastAppVersion: 45.2.0
     font.internaluseonly.changed: false
     font.name.sans-serif.x-western: Arial
     font.name.serif.x-western: Times New Roman
     gfx.blacklist.direct2d: 3
     gfx.blacklist.layers.direct3d9: 3
     mail.openMessageBehavior.version: 1
     mail.spotlight.enable: true
     mail.spotlight.firstRunDone: true
     mail.spotlight.global_reindex_time: 1337434466
     mailnews.database.global.datastore.id: 091ab813-5cff-9342-8076-c2054fd19e8
     network.cookie.cookieBehavior: 1
     network.cookie.lifetimePolicy: 2
     network.cookie.prefsMigrated: true
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1467090269
     places.history.expiration.transient_current_max_pages: 104858
     plugin.importedState: true
     plugin.state.adobepdfviewernpapi: 0
     privacy.donottrackheader.enabled: true

 Graphics

     Vendor ID: 0x1002
     Device ID: 0x9583
     WebGL Renderer: ATI Technologies Inc. -- ATI Radeon HD 2600 PRO OpenGL Engine -- 2.1 ATI-10.0.48
     GPU Accelerated Windows: 0

     AzureCanvasBackend: skia
     AzureSkiaAccelerated: 1
     AzureFallbackCanvasBackend: none
     AzureContentBackend: quartz

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 0
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.12
     4.12

     NSS
     3.21.1 Basic ECC
     3.21.1 Basic ECC

     NSS Util
     3.21.1
     3.21.1

     NSS SSL
     3.21.1 Basic ECC
     3.21.1 Basic ECC

     NSS S/MIME
     3.21.1 Basic ECC
     3.21.1 Basic ECC

With reference to the previous post....the "Trouble Shooting Information"....Account #5 is the one that is not allowing some mail messages to download. I don't know why there is a #5 when there is no #3 and 4. Account #1 is not functioning properly or I would use it. But, any time I select any of the folders on #1, I get a request for a password. When I enter a password (for instance my PW for AOL) or any other familiar PW I'm refused access to downloading.

One more thing. You might wonder why I have a verizon.net, address when I am on AOL mail server. I was with VZ when they dropped their mail service an turned it over to AOL, and I was assured I wud keep the same address. Now, as you may know, VZ Fios sold out their CA, FL, and TX operations to Frontier, so I am battling that problem also, with TV, Internet, and landline. tnx ~ gy

Actually I managed to make a fool of myself over the Verizon change to AOL, emailing everyone that it was Comcast. So I do understand where you are coming from.

Right click the account in the folder pane, select settings In synchronization and storage for the account, select keep messages for this account on this computer.

Thanks, Matt. Actually, my Old VZ account (a POP mail server) is where the prob is - I cannot get the AOL account to work at all because of the PW problem as stated earlier. So! The Old VZ account does not have the "Keep messages for this account on this computer. FWIW - it is selected on the AOL account.

With the POP account (Old VZ) there is not a category named Sync & Stor. On this account, the "Automatically download new messages" has been checked. "Leave messages on server" has been unchecked, as of now.

We'll have to wait for the next few emails to come in before we will know if this did some good. I'll' keep the forum posted on the results. BTW - the Mac app "Mail" has no problems, I just like the versatility of TB better. tnx ~gy

No Go! Received monthly news letter from my state rep, clicked to open it, the "Download" message showed up. I selected it and phoof! message is gone. I have no idea where it went. I tried pasting a screen shot of the message that shows in the message window, into this post but even tho I was able to capture the message, it wud not past here - understandable. ~ gy

A screen shot might really help here.. But as you observed you can not paste them. For help with creating a screenshot, please see How do I create a screenshot of my problem?

I tried uploading a screen shot but I don't see it here........I'm about ready to delete all mail accounts and/or remove TB and start over.....or, go back to Apple Mail. tnx Matt ~ gy

Oops! I guess it worked after all.....a bit micro for my eyes, tho.