There are many problems, not only in the lore of individual games, but also overall in Fallout that are incositent with the logic of the world we've been presented.

The first is, where are the bikes? We know for a fact there are running vehicles from Fallout 2 and New Vegas tells us there are planes. Once in and in Vertibirds it's clear vehicle technology isn't dead in this world. Electricity, Combustion, Steam Power, Batteries, Fusion, etc all work just fine. That said why aren't the simplest vehicle's; bikes, used? Even in Fallout 1 and 2 this is a flaw.

For example a horse can only carry 20 percent of it's body weight, even assuming brahmin weigh 2300 pounds, (And that's unrealistic considering most cows in todays world don't even weigh that), the most it could carry is 460 pounds and it has to constantly eat, get water, and it could easily be startled (as we see many times in game) from wasteland creatures. A bike outfitted to only carry supplies such as in South East Asia can carry 400 pounds and doesn't need food, water or get scared.

Further a bike would make the rider faster than even horses. Many test have found that bikes can go faster and longer distances than even horses! A bike rider could transverse 100 miles in a day, with 50 being the average if off road. That's much faster than 20 miles a day walking. Also in combat a bike would give any rider an edge and arguably against wasteland creatures if a bike rider were going downhill you could even outrun a Deathclaw. Considering horses supposedly don't exist anymore a bike would be of great benefit.

I could say more but for now I have to go. What logical problems do you see with the world of Fallout?

Rubber is far too precious to be wasting it on inner tubes and tyres. Especially when you have other methods of transport. Without rubber tyres, bikes loose all the advantages you've just claimed.