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How did I know that the road I was on was absolutely fed up?

Road as in "way for traffic"? Railroad? Road in any other sense?
Were you on that road? walking? driving? cycling? traveling?
Did it come to know for you that the road is fed up by something you saw? heard? smelled? felt? otherwise perceived?

Sundowner:

Road as in "way for traffic"? YES. Railroad? Road in any other sense?
Were you on that road? walking? driving? cycling? traveling? I was at an intersection of this road and another street. I happened to be driving, but it would work equally well for walking or cycling.
Did it come to know for you that the road is fed up by something you saw? This. heard? smelled? felt? otherwise perceived?

When you say "fed up," do you mean "having an attitude of being completely done with something"? Is there any literal feeding involved? Is wordplay involved?
Were any other people traversing the road? Crossing it? Any other creatures?

Did you see something? See someone? Read something? Was something broken? Occluded? Missing? Weather relevant? Road conditions? Was it an asphalt road? Relevant?

Balin:

When you say "fed up," do you mean "having an attitude of being completely done with something"? Yes. Is there any literal feeding involved? Is wordplay involved? Yes. But not in the definition of "fed up", just in the puzzle overall.
Were any other people traversing the road? Crossing it? Any other creatures? Not relevantly.

Galfisk:

Did you see something? Yes. See someone? Read something? This. I read the same thing I saw. Was something broken? Occluded? Missing? Weather relevant? Road conditions? No to all of these. Was it an asphalt road? Relevant? It was, but not relevant other than to show that it was, in my neck of the woods, a standard road for vehicular traffic.

Is the name of the road relevant? If so, does the fact of what the actual name is show in itself that the road is fed up? Or is there a mistake on, damage to, or some other similar alteration to one of the signs that changes the name, suggesting that the road was fed up enough with the original name to change it?

...alternately, given that there was an intersection, is the name of the cross street relevant? I'm thinking something like you were at the intersection of Sick Rd. and Tired St.

Is the name of the road relevant? Very much so.If so, does the fact of what the actual name is show in itself that the road is fed up? Yes. Or is there a mistake on, damage to, or some other similar alteration to one of the signs that changes the name, suggesting that the road was fed up enough with the original name to change it?

..alternately, given that there was an intersection, is the name of the cross street relevant? I'm thinking something like you were at the intersection of Sick Rd. and Tired St. Irrelevant.

"name of the road" -- does that include the specific road designation (e.g. "road," "street")? Its abbreviation (e.g. "RD," "ST")?