Hi,
While I am second to cunderw with the suggestion to either check browser settings or contact Support to give guys a chance to correct the problem if it is on the TestComplete's side, the plain answer to your question is : yes, this is possible.
Create the OnLogError error handler (see documentation for more details) and suppress the required error message within it.
> All our functions are run in a try/catch construction
While this may be a reasonable approach in development world, in the world of TestComplete this is seldom required.
The case is that try/catch handles only language runtime errors like division by zero, null-pointer access, etc.
But most problems in TestComplete tests are not runtime but 'logical' problems that are not a problem from the language runtime point of view but are problems from the point of view of actions over test objects. For example, process crash or absence of some window is not a runtime problem (all at all, TestComplete has no idea of what are you checking. It is quite possible that the goal of your test is to check that process crashes after some actions or that some window does not exist), but it is a logical problem because you are explicitly referring to some object and this means that the object must exist.
Note, that there are two ways to reference objects in TestComplete. I call them explicit and implicit.
Explicit reference is when you address object directly in your test code. For example: Sys.Process(...). Explicit reference means that you expect that the object must exists at this point and thus its absence is treated as a logical error and is reported to test log. Note one more time - this is a logical problem but not the runtime one and thus it is not handled by try/catch.
Implicit reference is when you try to get an object but it is possible that the object may not exist at this point. In this case you must use WaitXXX() method in your test code. For example: Sys.WaitProcess(...). .WaitXXX() methods do not log the error if the sought for object does not exist but instead they return an empty stub object with the .Exists property set to false, so that you can check in your test code whether or not the sought for object was found and act accordingly.