How can we tell how many exoplanets a star has? Part of this answer (discussing the moving center of mass of our solar system) explains that this movement is one way we know a star has planets:


  Bonus: We use this phenomenon to find planets outside the Solar System! If a distant star is observed to 'wobble' or oscillate about it's mean position, we can use that data to infer the presence of one or more exoplanets, and calculate their mass.


It makes sense to me that we could tell that a start has some planet(s) in orbit around it, based on how they affect the star's position. But is this movement really precise and predictable enough to pin down how many exoplanets, and how large they are? Can we produce a "map" of a distant solar system based on this movement? 

If needed, we can even restrict the discussion to planets that are "big enough" to detect. For example, when performing this exercise on our Sun, would we be able to accurately predict at least the four gas giants, even if we couldn't get Mercury?
What you are describing is a basic signal processing problem. The doppler shift that one observes is due to the motion of the star in the system around the system's centre of mass. The star will be influenced by the gravitational pull of each of the planets in that system, each of which exerts a gravitational pull that increases with the mass of the planet and decreases with the orbital radius of the planet.

The overall motion of the star will be the sum of the effects of all the planets. Importantly, the effect of each planet will have its own amplitude and will be periodic with a period equal to the orbital period of that planet.

Let's imagine that each planet is in a circular orbit (elliptical orbits are more complicated, but the principle is the same). Each planet would cause a circular motion in the star about the centre of mass of the planet-star pair, leading to an observable doppler signal which has the form of a sine wave with a period equal to the orbital period of the planet. The amplitude of that signal will increase with the mass of the planet and increases with decreasing orbital radius.

The overall signal is the sum over all the planets in the system. Fortunately the decomposition of this signal back into its individual components is a well-trodden problem in physics, electronics and many other fields and is known as Fourier Analysis. Whether you can successfully recover the original signals from each of the planets depends on how long you observed the system (ideally you want to observe for longer than the longest orbital period) and the amplitude of the signals compared with the noise in your observations.

In general it is easier to recover high-mass planets with short orbital periods and more difficult to recover low-mass planets with long orbital periods.

The image below might be helpful. It shows the track of the solar system centre of mass compared with centre of the Sun over a period of several decades. Notice how the Sun executes a complex trajectory (with respect to the solar system centre of mass) that is mainly caused by the orbit of Jupiter, but then there are smaller, superimposed, signals caused by the smaller planets. In principle, if you observed for longer than the period of Neptune and had a detector which gave perfect measurements, you could reconstruct how many planets there were in the solar system, what their orbital periods were (and then from Kepler's 3rd law, what the planet-star sepration was) and what their masses were (multiplied by the inclination of their orbits with respect to the line of sight of observation, which is generally an unknown in doppler measurements). 



In terms of what we could currently see if we observed the Sun as a star: basically we would (assuming we observed for 20 years) detect Jupiter quite easily with a doppler amplitude of about 13 m/s. We would also see that there was a drift in Jupiter's signal due to the influence of Saturn, but we would have to observe for >Saturn's orbital period in order to confirm the presence of Saturn, its orbital period and mass. The inner planets produce an amplitude that is too small to be visible using the technology currently available. e.g. The Earth would produce a doppler wobble of amplitude $&lt;8$ cm/s, but the current precision of doppler measurements is limited to about 50 cm/s.

The doppler amplitudes in m/s due to each of the planets (assuming we view them edge on) are:

Mercury &lt;0.01
Venus    0.08
Earth    0.08
Mars    &lt;0.01
Jupiter  12.5
Saturn   2.6
Uranus   0.28
Neptune  0.26

Thus with current technology, only Jupiter and Saturn are detectable.

Below I simulate what the doppler signal due to these two planets would be. I hope you can see that the overall signal consists of the superposition of two sinusoidal signals with different periods and amplitudes. Myriad computational tools are available to do the fourier decomposition to establish these.