Question: So I have this problem scratching my head. I have a number X (lets say 800) and I want to ensure that I can consecuentaly divide it by 2 four times and all the results will be %2==0.

800 fills the condition as:

$\frac{800}{2} = 400 % 2=0

$\frac{400}{2}$ = 200%2=0

$\frac{200}{2}$ = 100%2==0

But 1200 doesnt, as:

$\frac{1200}{2}$ = 600%2=0

$\frac{600}{2}$ = 350%2=0

$\frac{350}{2}$ = 175%2=1

Is there any quick way to ensure this condition will fill given a number X? If so, is there also any way to find the next number who will fill it?

Answer: 
For $N$ to have this property, $N$ must be a multiple of $8$. That's it!

You might say "But what about $1200$? **That's** a multiple of $8$ and it didn't work!" Well...it only "didn't work" because you said that $600/2$ was $350$ instead of $300$. The correct version is 

\begin{align}
&& 1200 \%2 &= 0\\
1200/2 &= 600; &600 \%2 &= 0 \\
600/2 &= 300; &300 \%2 &= 0 \\
300/2 &= 150;&150 \%2 &= 0,
\end{align}
just as expected.
Your pattern isn't working because you have a small calculating mistake. 600 ÷ 2 = 300!
For any number n, n can be divided by 2 four times, if n is a multiple of 8.