Google claimed there was suspicious search activity from my computer when it stopped letting me search altogether. Before that I had to enter a captcha more and more often in order for Google to answer my search request. I search only infrequently.
I scanned my PC - started the scan from a special anti-virus USB boot system so that the actual Windows 7 of the PC was not running, so no root kits or anything could influence the result. I ran two different anti-virus apps. None of them reported anything.
Since my router had not been supported since 2011 I thought maybe that device might have been hacked. So I got a new well-supported one.
All was clear, Google let me search again - but now, less than a month later, I just had to answer a captcha again before Google would let me search.
Does anybody have an idea what might be going on here? Unfortunately Google does not provide any specifics in their messages, neither do their help pages about this subject.
I also ran the Google Cleanup Tool and it reported nothing.
So I looked at my traffic with Wireshark. Unfortunately that's next to useless. I only care about traffic to 1e100.com - Google search servers. Well, there's plenty of such traffic - if Chrome is running. Even with the only open tab being an empty one, i.e. no page is loaded. Someone is chatting with Google servers, but it may very well be normal for Chrome, even though I wonder why it needs to chat with Google servers so much when there is nothing going on in the browser. Traffic analysis beyond the fact that communication takes places is not possible - it's all SSL. So I'm not the wiser.
EDIT^2: (25 September, 2015) I ended up buying a new router. Maybe my router got hacked - there have been no updates for the firmware for five years. For the months since replacing the router I have not seen the above issue.
UPDATE^3: Even with the new router I had this issue again - but only briefly (once). This helped me discover that I had just enabled a Chrome extension just before Google warned me. I disabled that particular extension again and Google never warned again. At the time I posted this question here I had the extension running too. The extension is BetterTTV - I add the name reluctantly as I don't have any proof and didn't examine this further, so others may want to look into this a bit more before raising your pitchforks against the extension. Noteworthy: The extension seems to load much or most of its code from the Internet. The author wrote this was to work around Google extension website adding new versions with too much of a delay.
Update^4: A few months later (Dec 2015) with the new router and without the mentioned Chrome extension, and I have not had the issue any more except for what I wrote in update #3.