Perfectly sensible question. I think the important thing here is to consider the size of the anomaly not relative to the speed these craft are travelling but the precision that we can predict & measure their expected speed.
So we're left in a position where there's a difference between the prediction and the measurement, when the measurement is highly accurate. That means something is out, and that might be extremely important for long range flights.
I'm not too sure how to work some of these things out, but it's possible that a few mm/second difference really adds up when you're on an enormous trip. If you're out by 10mm/second here, then your next flyby will leave you even further out, then the next, etc. until you crash into a planet. Cassini was out by under 1 mm/s, but its flight path would have been 2 billion miles. These things travel for years, and can't make too many corrections.
Another one of those questions: doesn't the acceleration of such a fly-by depend in large amount on the masses of the various objects, and if that's correct could this indicate an error in the mass of one of the objects (say, Cassini having a mass (not a weight) 1/1000000 less than we think it does)? And also, could it be indicative of a number of factors all being off by a fraction of the resulting discrepancy, or even two factors where one is a negative and the other is a slightly different positive?
The mass of the "test particle" Cassini will cancel out of orbit equation but will matter for drag, radiation pressure etc. If it's outside the error bars they gave for mass, then this is conceivable.
I think the measurement aspect is only part of the interest. The other is there might be previously unknown physics at work. We don't get too many chances to experiment on things in space, so surprises mean there's an understanding gap.
At one point, the motion of Mercury didn't fit into the rules of mechanics, which we thought we understood. It turns out we didn't, and general relativity filled that gap.
It's far more likely the explanation is a mundane force not considered before, than new physics. But we have to look.
So we're left in a position where there's a difference between the prediction and the measurement, when the measurement is highly accurate. That means something is out, and that might be extremely important for long range flights.
I'm not too sure how to work some of these things out, but it's possible that a few mm/second difference really adds up when you're on an enormous trip. If you're out by 10mm/second here, then your next flyby will leave you even further out, then the next, etc. until you crash into a planet. Cassini was out by under 1 mm/s, but its flight path would have been 2 billion miles. These things travel for years, and can't make too many corrections.
More info on the crazy journey it took: http://www.theplanetstoday.com/cassini_flight_path.html