It is possible that
Your computer is connecting on an outside wireless network that is on a different router.
Port is blocked in Firewall (port 5003)
The FMP installation needs to be reinstalled.
The file itself may not have sharing turned on or it may have sharing enabled but not for the privilege set you're using on the other computer, or it may have it turned on but not set up for that specific database file. To check it, go to the host box and go to File menu --> Sharing --> FileMaker Network. At the top Network Sharing should be "On"; on the lower right Network access to file, under File: "yourfilename.fp7", it shoudl either say "All users" or "Specify users by privilete set" and then specify the priv set that will be used on the FmPro 8.5 box. Oh, and make sure "Don't display in Open Remote File dialog" is NOT CHECKED.
Also: If you have any kind of firewall it may be blocking.
Easily Launch a Remote FileMaker Database from a Desktop Icon
How to access FileMaker from behind a Firewall.
How to login to a FileMaker database from home or on the road from a users perspective.
No answers here
Why are files that I used to have access to via the Open Remote dialog within FileMaker Pro, no longer available?
Answer ID: 9362Last Updated: Feb 16, 2012
Opening a Shared File in FileMaker Pro
Answer ID: 9827Last Updated: Oct 04, 2011
apologies to David if I am repeating an answer, but since I dont' see it in his list....
You may have a missing or damaged .PEM file.
For More Information see: Can't login to FMS database on new Mac, other Macs can
This is one of many acknowledged bugs that can be found in the Known Bug List thread here in the Report an Issue section of the forum.
It can also be downloaded as a database file from: https://www.dropbox.com/s/jt09b82i0xijbu3/FMP%20Bugs.zip