The essential flaw in this reasoning is that you think that time is a fixed feature in the universe. There is no such thing as "a specific time". A "specific time" would have to be measured by someone and their measurements would not agree with any other measurements made by anybody who was travelling at a different relative speed. If time is not fixed then speed is not fixed either, speed being the measure of distance travelled over time taken. To put it simply, in your OWN inertial frame of reference you can never move at or close to the speed of light. To you, light will always appear to travel at 3.0 x 10^8 m/s. Towards you, away from you, it doesn't matter. What will change is your perception of time compared to somebody who is moving at a different speed, relative to yourself.
The important thing to think about is that when you say this observer is moving close to the speed of light, you must specify what this speed is RELATIVE to. Relative to the Earth, the sun? Relative to Observer B? Speeds are always relative to something, and two observers moving relative to one another will measure the speeds of non-light objects to be different because they perceive time differently. Light, however, behaves in a different manner from this. Two observers in different frames of reference will always measure the same speed of light relative to themselves but if they meet later they will either disagree about how long the event took or the distance involved.
Example:
The observer A moving at any constant speed (his inertial frame of reference) will see the light pass him at the speed of light relative to his own speed, whatever that may be, and it will take the appropriate amount of time (to Mr. A) for the light to strike the object. Let's say Mr. A measures this amount of time to be 1 second, using his watch.
Observer B, standing to the side and moving at a different, but constant, speed (his inertial frame of reference) watches as Mr. A passes by him and measures that A is moving at close to the speed of light, compared to himself. He also sees the beam of light pass by him and measures that the beam of light is moving (of course) at the speed of light. He then checks his watch and sees that the light took 30 seconds to strike the object.
So, observer A and observer B now have the same measurements for the speed of light, but they disagree about how long the event took to happen. Who's right? Both of them are. The difference is that their measurements of time and distance will not agree, even though their measurements of the speed of the light beam are the same. Since speed = distance/time then simple math can show that you can have equal measurements for speed when your time is different ONLY if the distance travelled is also different. What this means is that while the observers agree with how fast the light was moving compared to themselves, they would both measure a different DISTANCE that the light travelled. This is length contraction.
On the other hand, if both observers would agree that the distance was the same (as is likely), then you would get the same result from the equasion only by showing that the TIME they measured was different, as in the example above with speedy Mr. A and pokey Mr. B. This is time dilation.
The practical upshot of all this is that ALL measurements of time and distance are PERSONAL measurements only. 1 second to me and 1 second to you are only the same if we both occupy the same inertial reference frame (i.e. if we're both travelling in the same direction and at the same speed). As noted in another thread, even a clock at the bottom of a mountain will show a different progression of time than a clock at the top of the mountain. Time passes differently at the top compared to the bottom and observers making measurements of speed, distance and duration using these clocks would find that none of their measurements would exactly line up EXCEPT for their measurement of the speed of light.
I hope that this makes some things clear as to why none of these theoretical observers would ever measure the speed of light to be anything but "c", compared to themselves. Time and space will bend to make it so.
*Edited for clarity*
