Thanks for your helpful answers.

It may or may not be a problem depending on the application and its operating environment. Using a single point of connection is a good idea if you want to avoid setting up ground loops (which will pick up AC mains interference). So for your scenario each PCB should be electrically connected at one point to the chassis, leaving the other points of attachment for each board as mechanical connections only.

In an electrical system, a ground loop refers to a current, almost always unwanted, in a conductor connecting two points that are supposed to be at the same potential, often ground, but are actually at different potentials. 

With two or more points connected to ground you can have some measurement problem. 

In order to avoid emc problems, there are some rules you must consider. First, you must estimate the spectrum of the signals involved and to consequentially consider the relative propagations and couplings; then, you must avoid to form any loops of current with meaningful areas; you must use short connections; You must pay attention to the ground connections: the currents return to their source on the paths of lowest impedance.

You know your case; so you can find the answer by looking the chapter 11 in the book od Clayton R. Paul, “Introduction to Electromagnetic Compatibility” second edition, A John Wiley & Sons, Inc. Publication.

Most of common answers you have already got. But if you need another theoretically-grounded view on what happens if you attach your PCB to chasis on multiple points please consider chapter 9.9 Chasis Connections ('chasis stitching') in the book Elya B. Joffe and Kai Sang Lock "Grounds for Grounding", IEEE Press and A John Wiley & Sons, Inc. Publication.

Thanks for your helpful answers.

Actually, it can be beneficial to connect the PCB signal ground plane or sheilding plane to chassis at mutiple points. Why? The same reason signal reference grids are used in data centers, etc. The parallel connections reduce the self inductance of the path for return currents that are seeking chassis or power system ground or the facility grounding electrode system.

Using a single chassis connection does provide a "bird on a live wire" scenario a.k.a. single point ground. However, that is commonly defeated by signal ground connections to other PCB's or external equipment. Its very difficult to maintain a "single point ground" in any integrated system with multiple metallic signal ground paths either to other PCB's or external equipment, e.g. Kirchhoff's Junction Law.

https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws

I would also suggest IEEE Std. 1100-2005 as many PCB and equipment designers or engineers don't have a firm grasp on the external factors relative to power and grounding of electronic equipment, e.g. the first step is understanding the electrical environment and useage of the equipment.

https://standards.ieee.org/findstds/standard/1100-2005.html

Myself, I prefer 4 connections to chassis, e.g. one at each corner of the PCB, typically this will be the shortest path taken by any unwanted EMC currents no matter where they are injected into the circuits. It may be desirable to have additional connections to reduce the loop areas. Also keep in mind every conductive path has self inductance which can greatly affect the impedance, especially with transient events. Sometimes we forget the basics, like  E=L(di/dt).  Even the mounting post and screws play a role. A single chassis connection is asking for trouble if the connection is not properly torqued. Having 4 is an insurance policy in this respect.

Also, think about it this way; if the L (self inductance) in the equation above is reduced by a factor of 4 by using 4 mounting post connections, then any voltage potential between the PCB signal ground plane and chassis is likewise reduced by a factor of 4.

 Lord Kelvin: “When you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers your knowledge is of meagre and unsatisfactory kind; it may be the beginning of knowledge but you have scarcely progressed in your thoughts to the stage of science whatever the matter may be.”

To that end, I offer you some additional resources. I started off my exploration of EMC with "Electrical Interference" by Rocco f. Ficchi (circa 1964) He also wrote a second book circa 1971. However, I also recommend Henry W. Ott's books. His webpage has lots of additional useful resource listings.

http://www.hottconsultants.com/book.html

Al