Does bottle water rise a little bit on full moon days? 
High tides and low tides are caused by the Moon. The Moon's
gravitational pull generates something called the tidal force. The
tidal force causes Earth—and its water—to bulge out on the side
closest to the Moon and the side farthest from the Moon. ... When
you're in one of the bulges, you experience a high tide.

If ocean water rises on full moon. And gravitational acceleration is not dependent on the mass of the attracted body. Just as a metal ball and feather falls at the same speed, why doesn't both bottle water and ocean water rise by same levels on a full moon?
If air is the reason, then on an atmosphere less planet does bottle water and ocean water rise by same levels?
 A: Does bottle water rise a little bit on full moon days?
No.  Tidal forces are about the difference in gravitational pull at different points in the same body.  For oceans and other very large bodies of water, this difference causes water to flow from one region to another, which causes the rise in tides.
For example, this is why, even though the sun's gravitational pull is much larger on the earth than the moon's, the moon dominates the tides because it is closer to the earth and therefore the difference in gravitational pull is larger.
So for the bottle, the difference in gravitational pull from one side of the bottle to the other side of the bottle is extremely small because the distance is extremely small relative to the distance to the moon, and the tidal forces can not be observed.
A: 
So for the bottle, the difference in gravitational pull from one side of the bottle to the other side of the bottle is extremely small because the distance is extremely small relative to the distance to the moon, and the tidal forces can not be observed.

How small? Let's work it out. The acceleration on a bottle of water due to the Moon is...
$$\text{Gravitational constant} \times \text{mass of the Moon} \times \frac{\text{diameter of the bottle}}{\text{distance to the Moon}^3}$$
Let's assume our bottle has a diameter of about 0.06m. The distance to the Moon varies, I'll use the semi-major axis. It won't make a difference.
$$\text{mass of the Moon} = 7.34 \times 10^{22} kg$$
$$\text{distance to the Moon} = 3.84 \times 10^8 m$$
$$\text{Gravitational constant} = 6.674 \times 10^{−11} m^3⋅kg^{−1}⋅s^{−2}$$
Which is about $6 \times 10^{-15} m⋅s^{−2}$. This is an order of magnitude smaller than the smallest acceleration measured. Not just extremely small, unmeasurably small.
A: As correctly stated in the deleted answer, the tides have nothing to do with the fact that the moon is full. Tides occur always, regardless of the moon being full or not.
Imagine your bottle filled with water is free-falling towards the moon, the bottle parallel to the moon's gravity field. What will happen to the water inside? On one side of the bottle, the gravity field will be higher than on the other end. Though very, very little (this isn't the case for a large body of water like that of the oceans). This difference in the gravitational force is called the tidal force. I think you can see that this tidal force is easily overcome by the force which holds the water level inside your bottle fixed. The water level increases a tiny, tiny bit, up to the point where the molecules in the water are separated enough to counteract the differences in the gravitational field. Only when the gravitational field is high enough, then the level will visibly change. The water will be stretched into droplets, and maybe into individual molecules. And the molecules? Think about it...
A: Even the most rewarded answer here has missed out on the fact that the entire bottle of water will rise by up to a meter due to the phenomenon of earth tides. The body of earth
is deformed by the same lunar and solar tidal forces that cause the sea tides. See https://en.wikipedia.org/wiki/Earth_tide.
A: A less known fact about the tides is that the lithosphere has few centimeters of tides, too.
So yes, your bottle as a whole will rise that high two times a day.
A: Since water is compressible it will expand when there is less gravitational pull towards the center of the Earth, which is the case on the Moon-facing side of the Earth. It's just not very much because water is not particularly well compressible.
The water surface in the bottle will also, like the oceans in the simplified tide model, have a bulge — a minuscule deviation from the curvature it has anyway because of the inhomogeneous gravitational field of Earth.
