It appears that advice from The Lady from Philadelphia is again needed.

The Lady from Philadelphia uses the splitter supplied by Comcast. One of its
outputs connects directly to her Motorola modem, using a double-male
(hmmm...) barrel connector from MCM. The other output connects to a cable
that feeds her Vizio monitor. She has no problems.

If changing the splitter doesn't fix the problem, then the problem is
probably //not// with the splitter. (The Lady from Philadelphia does not
understand why this is not obvious.) Furthermore, it is not clear to her how
soldering the leads is going to fix //anything//. How does one //get// to
the leads, anyway? F connectors were not designed for soldering to their
innermost... parts.

The Lady from Philadelphia's gut feeling (pardon her vulgarity) is that your
modem has a bad F connector, making intermittent contact. She suggests that
you drop by the cable company and explain the problem. Of course, as you are
(likely) the modem's owner, the technically ignorant people behind the desk
will probably tell you that you need to buy a new modem. Perhaps a veiled
threat (of your choosing) will encourage them to loan you a spare, so you
can see what happens. Do not forget to have your account's MAC changed
before you leave, or your Internet cable connection will be deader than
Marley.

There is also the possibilty that the little bare wire sticking out of the
cable is, shall we say, too short to perform its required duties? Have you
experimented with a better-endowed cable?

The Lady from Philadelphia concludes by offering this rude (and unsolicited)
observation... TWENTY BUCKS FOR A BLEEPING SPLITTER? She hopes it was at
least gold-plated. The Lady from Philadelphia knows from personal experience
that expensive items from RodentShack (polite folk do not talk of r**s in
public) are not necessarily better than those from elsewhere, costing
one-fifth as much.