What do you mean by "led to believe"? What part of Star Trek suggests that the Federation should be wider than three warp-months?
I'll use
Guy's method of providing quotes, since that seems to be the way to convince people of things:
ST: First Contact said:
LILY: How many planets are in this Federation?
PICARD: Over one hundred and fifty ...spread across eight thousand light years.
LILY: You mustn't get home much.
DS9 The Sound of Her Voice said:
BASHIR: We need more speed.
O'BRIEN: Speed's not the problem. I could increase the warp plasma ninety seven gigahertz. That would increase our velocity to warp nine point five and save us almost a full day.
WORF: The problem on the Defiant is how to maintain structural integrity when we go above warp nine.
O'BRIEN: Exactly. At those kinds of speed the ship literally starts tearing herself apart.
SISKO: Is there any way to strengthen the structural integrity field?
O'BRIEN: Not without bleeding power from some other source.
TNG Bloodlines said:
DATA: The ship is holding position approximately three hundred billion kilometres from here.
PICARD: Plot a course. Maximum warp.
RIKER: Even at warp nine we wouldn't get there for another twenty minutes.
Using maths (yay!) we determine that 300 billion km is 0.032 light years in 20 minutes, which at that speed (if maintainable) is around 2.2 light years per day at warp 9.
Using the formula from the TNG bible,
speed = wf^(10/3)*c, warp 9 would achieve a speed of 4.1 light years per day (I've already found an instance of continuity error, and I haven't even gotten to the crux of my analysis

)
So, let's assume 4.1 light years per day is the velocity at which you can attain if you maintain constant warp power (also unlikely). This may be reasonable for an Intrepid-class ship, but for the Defiant-class USS Valiant, it's not. Even if it were:
Traveling 8000 LY at warp 9 non-stop would take a Defiant-class ship nearly 2000 days, or 5.34 years. This assumes that a Federation that is "spread across 8000 light years" is a circumference, and not a diameter. If it is a diameter, now you're talking 25,000 LY (
pi*D), and a travel time of 16.79 years. And that's assuming you started/ended at a point on the outer edge, and not from Earth!
Meanwhile, at warp 9, a Defiant-class ship can only travel a distance of 369 LY in a period of 90 days.
In any rate, circumnavigating the Federation in a period of three months is impossible. It's extremely unlikely that even a cursory overview of the Federation could be accomplished.
So, my original thesis that there is a continuity problem in Star Trek stands.