Can I play?

There are ways, depending upon how the Readers/clients and servers are
configured.

IF the downstairs computer is set to automatically poll [timed retrieval]
the providers for new mail AND delete the copy on the server, then it would
appear that both upstairs and downstairs would be able to retrieve the same
mail IF the upstairs computer happens to access the mail first AND is not
set to remove it, OR if the timing just happens to coincide [manual check
upstairs, downstairs accessing at approximately the same time].

On the other hand, IF the upstairs computer is set to download and remove
those messages, then depending upon WHEN the downstairs computer polled the
servers, the mail may or may not be available to it. MailWasher likely would
not have run on either UNLESS configured to do so AND/OR was running when
this occurred, or is not receiving the mail once the server marks it as
accessed/downloaded/read.

Other potentials to be taken under consideration would appear to be:

Whether the Pop3, hotmail, IMAP, or other servers, had been set to
automatically mark messages as READ once accessed [read messages MAY not be
downloaded as they are not NEW (un-accessed) on the server] though they may
remain on the server, so if neither of the computers are set to remove
copies from the server or the server doesn't allow this, the mail might be
either available or not, depending upon how the actual mail service and
clients are setup {all messages vs. new messages vs. marked/not marked vs.
synchronization vs. first access vs. removed/moved or not}.

Whether the messages are actually downloaded or are just being viewed in
the preview pane [online/direct access verses retrieval and offline READ]
and the MARK AS READ settings in conjunction (server and client) with the
previously mentioned aspects AND synchronization settings [mail is not
marked READ on the server AND/OR mail copy not removed from server UNTIL
read} until next sync [offline verses online or new download, at which time
this occurs for READ, deleted, moved, and sent messages, again depending
upon client and server settings].

As you previously had phone access, these settings would not have been so
important, but as you now apparently access the NET via on
access/request/always on (per your other postings), these settings should be
checked and re-configured for such access. Or are you still phone line?

Additionally, if your ISP is configured to download directly from your
services or one of your services/servers is [one account collects from other
accounts], it may be set to NOT synchronize/remove/mark as READ/ACCESSED,
until one of the clients sends that data back to the ISP/service {which then
transfers that back to the actual mail/news servers}. So pulling messages in
this fashion MAY allow multiple views {and apparent message loss} IF one
client collects messages directly and the other client collects messages via
the collected source once the servers are synchronized.

But then what do I know, I just pulled this from some dark depth of my
brain as seemingly reasonable, this may be just a bunch of  letters on your
screen ... ;Q

Open Outlook Express on the downstairs machine.
Select Tools - Options
Is "Check for new mail every (30) minutes" ticked?