I had a similar issue come up a long time ago, but I thought it was an isolated incident. I recently discovered that it's not isolated at all! This is the kind of thing I refer to as an "innocent mistake" (or misunderstanding) that could get a conversion candidate into significant trouble. This is precisely the kind of stupid mistake that a bully could latch on to, if you have a bully or suspect one. So if you can avoid these issues, do so. If not, be prepared to defend yourself. And if you're lucky, your rabbis are so technologically clueless that it never comes up at all (or it just doesn't happen).
Quite frankly, I think most of these problems stem from the fact that rabbis are generally not the most computer literate people on this earth (and generational differences can exacerbate this knowledge/use gap). Rabbis may not understand how computers or the internet work and there's always the possibility of a technological gaff being exploited by a bully if someone's got it out for you.
On the other side: Rabbis, you need to chill out. A computer that appears online during Shabbat really might not be the person online. Really.
So let's discuss how you might be "online" when you're not really online. I call this passive internet use.
The most common: you just forgot to turn off the computer before Shabbat began because, as we all know, the Shabbos Shuffle is frantic and there is always something forgotten. If your computer remains on and online, your chat program could continue to be on during Shabbat. I see this a lot with Gchat accounts-all that needs to be open is the email. Many privacy settings in chat programs allow you to disable the "idle" status in order to keep people from stalking when you're at the computer or not. The "idle" status shows other chatters that you have not used your computer/the program within a certain amount of time, usually 20 minutes to an hour. So if your program can allow "idle" statuses and you aren't marked as "idle," you must actually be in the chair, right? Some chat programs, like Facebook, don't even have an "idle" status, so people just always assume you're there unless you don't respond. (An alternative fix: block the rabbi, bully, or others. Then they'll never see you online. Remember that Gchat automatically adds people you email with to your chat list, so you might never have "friended" these people!)
Pre-programmed or automatic postings. For example, I use Hootsuite, which can program automatic postings to go out on your Twitter or Facebook account at a specified time. Someone who works in social media or other internet-based work might benefit from these automatic postings keeping them visible in the internet sphere or maintaining their Klout score. As most of the observant Twittersphere knows, Klout scores go down for all of us every week after Shabbat, sometimes significantly. We didn't tweet for 25 hours while everyone else was tweeting, so we must not be as influential as those people! Another example is what I do with this blog: all my posts are pre-written and automated to post at a specific date and time. And heaven knows I get an earful if a post accidentally posts on Shabbat because I wrote the wrong date. (That's happened on this blog twice. As you can imagine, the people who jumped on me about it were computer-illiterate and just plain rude.) Auto-posting isn't an option I choose to use on Shabbat or yom tov, but they do exist and someone who needs a bigger online presence than I do might feel the need to take advantage of these options. However, if this is you, I suggest adding "[auto-tweet]" or other clarification language to your automated postings to avoid this mess in the first place. (This also protects you from later comments from an employer like, "But I see you work on Saturdays all the time!" This could be a real problem and could get you fired, maybe even negating any religious discrimination wrongful termination claim. I don't know, but that seems reasonable.)
Emails that say they were sent during Shabbat. This one is a bit beyond my technological knowledge, but my understanding is that it's very possible for emails to have an incorrect "time sent." If you have this happen to you, saying you sent an email during Shabbat, and it happens to be to your rabbi or beit din, then you know that Hashem has decided you need a test. I don't envy that, but it apparently happened to someone. I do know that when your email has a problem reaching its destination, your email will continue "trying to send" the email periodically for up to several days. This could certainly happen on Shabbat, but you would have proof that the email server did it automatically since every email server I've used sends a "your email failed" email, along with follow-ups for each unsuccessful (or not) future attempt. Try to read the times on those and maybe not delete those emails for a few weeks if you could see this being a problem.
Anti-virus programs, other automatic updates, or software installs. Someone very smartly said that they run their anti-virus software on Shabbat because they don't need the computer for anything else. This goes along with the "leaving the computer on" issue above, but apparently this could create a separate issue by bringing a computer online that was offline before. This is a particularly smart productivity idea if you make your living on your computer. Computer scans can take hours, reducing your computer's speed (and thus, usability) during that time. So if you can move that less-productive time to a time period when you don't need to use the computer, bam! You've just made a few hours of your life more productive! Less regularly, you may do the same for new software or other major updates.
These are the issues I can think of or have been made aware of. Do you know of other ways your computer might be passively online during Shabbat? If so, note them in the comments below.
Let me be clear here. I don't think it's a bad thing for rabbis to ask a conversion candidate if something like this happens. I think the question should be asked. But if there is a reasonable explanation (or clueless confusion), there is no reason to blackball a conversion candidate for it. There are simply too many reasonable explanations for it.
A related issue: being told you're online "too much." This is more unusual, but that's actually the scolding I got. (From not one source, but two! I'm that lucky.) This is more likely the further to the right you go. Besides the fact that it is super creepy that this person is monitoring your time online, you are rightfully entitled to tell them that is irrelevant and not their business. Of course, say this as carefully as you can because you don't want to create bad feelings. But honestly, it's not their business (unless you live in a no-internet-allowed community, and then that's what you signed up for). This goes over much better if you can say that you have a significant computer-based workload, for instance as a freelancer, blogger, web designer, student, whatever. I think part of this comes from a lack of understanding of computer-based work and that all you have to do is leave a browser on in the background for them to get this impression that you are mindlessly surfing the internet for 17 hours a day. Blocking the Nosy Nancies is also a useful technique here.