Pull a bulb, take an volt meter and measure from the prongs inside to the neg terminal on the battery, with the bike started one of them should have +13 volts or so across it. If not then you likely have a open wire. Either way its easy enough to workaround if your not getting current to the taillight running lights, you can tap into another wire thats hot while the bike is on and wire your tailight running light wires direct to that, can get away with doing that as long as the brake brights are working fine, which if they are by the way, means your open wire is a positive, not your ground since the bright and running light filaments share a ground. Personally I'd grab my multimeter and trace the wire back to wherever its losing power and just fix it properly. On the other hand, if you DO find a +13 or so voltage when you test where the bulbs plug in then its your bulbs or the housing they go in. Of course I've never looked at a wiring diagram for a 99 f4 specifically so what do I know, lol.

 

Actually, just looked at a wiring diagram for your bike, and theres a ton of things running off the black with brown stripe wire thats powering your tail light running lights, including your brake brights, through a switch. So if your brake brights work then that wire is powered, and your ground is good, so just traceback the black with brown stripe wire from your tail lights till you find where it still has power in it. Its the same wire thats the power input on your brake light switch, so somewhere between the two points theres a break in the line. Otherwise its either the housing that holds the bulbs are corroded or broke, or you got the wrong bulbs.