Yes, and I use the combination all the time. Assuming this isn't something trivial such as a dead battery...
Attach the ST-E3-RT to the hot shoe on your 5D III.
Power on the 5D III
Power on the ST-E3-RT transmitter
Power on the 600EX-RT flash
On the back of the 600EX-RT flash there's a "link" button. It looks like a zig-zag arrow. It's the left-most button the back of the flash (just above the button labeled "mode"). This button cycles through various linking modes. But there is only ONE mode that works with the ST-E3-RT.
The ST-E3-RT only works in "radio" mode and it only works as a "master" (it would be pointless to be a slave since it doesn't actually have a flash).
The 600EX-RT, on the other hand, can work standalone, or it can wirelessly in either "optical" mode or "radio" mode. And it can be a "master" or it can be a "slave. You MUST configure it to work wireless in RADIO mode and it must be a SLAVE.
As you press the zig-zag arrow button (the link mode button) you'll see the LCD display on the back of the 600EX-RT display an icon in the extreme upper right corner. You want the thing that looks like a lollipop with waves emanating from it (that's "radio" mode) AND you want to see the word "SLAVE" on the LCD display.
When you reach the correct mode, IF the ST-E3-RT is on AND the 600EX-RT is in the correct mode THEN you'll get a green "LINK" light on both units (meaning they've established radio communication with each other.)
There can be several reasons why it wont work.
Press the right-most soft-button to get to the page labeled "Menu 3". On that page, the first soft button will be labeled "CH" (channel) and the next soft-button is labeled "ID" (identification number). These two options allow you to work in places where other shooters are using the same gear without THEIR cameras firing YOUR flashes (and vice versa.)
So you do have to make sure your units are all using the same channel. By default this is set to "auto" and that's probably fine.
But even if units are on the same channel, they'll only agree to work together if they use the same ID. I think the factory default is to use ID "0000" (all zeros) but you could set it to some other value. E.g. if you picked "1234" then EVERY COMPONENT must use the SAME ID (they all have to be set to "1234" or they wont link and talk to each other).
Next... it's possible to assign flashes to firing groups. By default your camera will use a firing group named "ALL" which, as the name implies, fires all flashes (as long as they're part of your wireless flash setup -- meaning they're on the same "ID" number and using the same channel (if you manual set channels) or using "auto" if you've set everything to "auto" channel selection.
Otherwise you can assign flashes to belong to group A, B, C, D, or E (five different groups). This allows you to control the power level or ratio of light provided by the systems in each group. So I can set flashes to fire from the right, others from the left, and others to light a background, and still others to fire from behind but create a nice shine on the subject's hair (a hair light) and I could assign them to groups A, B, C, and D. I can then independently set how much power should be provided by each flash. You can put more than one flash into the "group" (which is why it's called a group).
Anyway... if you assigned a flash to group B but told the camera that you only want to fire group A then the flash in group B wont fire even though it's part of your flash system and is otherwise "working" correctly because you told it not to fire.
Syl Arena has a few books, blogs, and videos on the Canon Speedlite system. You can find a recorded webinar that goes in-depth on this system here:
Canon DLC: Gallery: Mastering the Speedlite 600EX-RT with Syl Arena
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