Does increasing RNG seed by 1 each time ensure not getting same consecutive values?

I've had a chance to see some interesting piece of code that was either used as an April Fools joke (update became public on April 1st) or was just a bug, because someone did not understand how to use RNGs.

Question is related to Random class being part of .NET/C#, but maybe other RNGs work the same way.

Simplified version of the code I found, after taking out all the unnecessary details, would look like this:

for ( int i = startI; i < stopI; ++i ) {
    int newValue = new Random( i ).Next( 0, 3 ); // new RNG with seed i generates a single value from 3 options: 0, 1 and 2
    // rest of code
}

I did run simple test of that code in LINQPad to see if what I was observing in program was just my "luck" or whether maybe that's actually how RNG used this way will work. Here's the code:

int lastChoice = -1;
int streakLength = -1;

for ( int i = 0; i < 100000000; ++i ) {
    int newChoice = new Random( i ).Next( 0, 3 );
    if ( newChoice == lastChoice ) {
        streakLength++;
        ( i + ";" + lastChoice + ";" + streakLength ).Dump();
    } else {
        lastChoice = newChoice;
        streakLength = 1;
    }
}

"The End".Dump();

(The Dump() method simply prints the value to the screen)

The result of running this "script" was just "The End", nothing more. It means, that for 100M cycles of generating a random value, not a single time was it able to generate same consecutive values, when having only 3 of them as an option.

So back to my question from the title - does increasing the seed of RNG (specifically the .NET/C#'s Random class, but general answer is also welcome) by one after every (integer) random number generation would ensure that no repeated consecutive values would occur? Or is that just pure luck?