Say I have a settlement two roads away from an intersection, but have another settlement only one road away from the same intersection and no other settlements near me. Can I build a settlement at the intersection?
Can I build a settlement at an intersection two roads away from one settlement and one road away from another?
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No, it is not possible
As best I can tell, this is the situation you are describing:
The 5th Edition rules say this about settlements:
Take special note of the “Distance Rule”: you may only build a settlement at an intersection if all 3 of the adjacent intersections are vacant (i.e., none are occupied by any settlements or cities—even yours).
In this situation, the intersection to the left is not vacant because it contains one of your settlements. Therefore, it is not permitted to build a settlement at this intersection.
Some players rephrase this rule as "a settlement must be at least two roads away from every other settlement". If that makes more sense to you, feel free to think that way.