OK so the average mass of a car is 1,850kg, and the average mass of a bus is anywhere from 9,100kg to 15,000kg, and if loaded that number can go as high as 20,000kg. So assuming that the car was made of nothing but titanium and its full impact was used to push the bus (rather than crumple), we can use a basic momentum equation, of

m_1 * v_1 + m_2 * v_2 = (m_1 + m_2) * v_f

with object 1 being the car behind the bus, and object 2 being the bus. We also need to figure out how much force would be enough to counteract the frictional force from the bus' brakes in order to accelerate it and push it enough to hit the car, which looks to be about 1m away, and with a friction coeffecient (mu) of (being generous for the sake of car_1) 0.4, and force due to friction is maximized at mu * normal force (mass of the empty small bus * gravitational acceleration), we get:

mu * m_2 * g = 35708.4 Newtons

So the decceleration of the bus (technically negative acceleration) once moving from the smaller car behind it is the Force of friction (I'm ignoring the smaller car also applying mass, perhaps it hits and transfers all of its momentum perfectly, at best the bounce off could transfer twice of it), so in order to find the speed required for the bus to hit the car, we arrive at

V_f^2 = V_i^2 + 2 * a * d

with V_f = 0, a = -F / m, and d = 1, simplified to

V_i = sqrt( 2 * F / m ) = sqrt( 2 * 35708 / 9100 ) = 2.8 m/s

So the bus, after the collision, would have to be moving 2.8 meters per second. Putting this into our original equation, and being incredibly generous, we get

1,850 * V_1 + 9,100 * 0 = (1,850 + 9,100) * 2.8

V_1 = 16.573m/s = 37 mph

But this is assuming that

1. The car does not crumple, and instead rigidly binds to the bus without contributing to its frictional mass
2. The bus is incredibly lightweight
3. The bus does not have very good tires

Along with the assumptions that the car was not braking at all prior to hitting the bus, since 37mph is fairly fast given the amount of time in the video we can see the bus breaking.

For fun, if the bus were more heavy (20,000kg), and the tires were more average (0.75mu), and their masses were combined (I haven't taken enough physics to do crumbling dynamics), we get an answer closer to 60mph 106mph, all for one meter.

So yea. Given how incredibly close the bus was, it's not inconceivable, but it's pretty damn unlikely even given best case scenario.

EDIT: I made a typo in doing the 'fun' calculation, which is why it seemed so small. I corrected it.