There is a sense in which you can't be too skeptical. Even when people tell the truth to the best of their knowledge, they might still be mistaken. People also tell deliberate lies. No source of information is completely reliable, and it's fair to say that all knowledge is tentative and subject to correction.
But there is, of course, a difference between that sort of skepticism and just flatly assuming something to be false because you think there's some reason to think it might be false or because you just don't like the source. From experience I'm skeptical of Fox News, but that doesn't mean all, or even most, of what they report is false. They even managed to get the Hello Kitty story right (about an erroneous claim that Hello Kitty was supposed to be a little girl rather than an anthropomorphic cat) when other media outlets got it wrong. See this previous post, in which I wrote, "Amazingly, Fox News managed to report the story accurately, even though generally speaking Fox News is neither a fox nor actual news. (I'm not sure what it is, but given the mix of on-screen talent, it's possibly a sugar daddy dating site.)"
I hate to sound like an old guy talking about how bad things are now compared with the good old days (though I'm admittedly old enough to get senior discounts without asking), but lately it does seem to me that people may be even worse than they used to be about rejecting information just because they don't want to believe it. The obvious example is Donald Trump's repeatedly labeling legitimate news sources as "fake news" whenever he doesn't like what they report and the tendency of many of his supporters to believe him in the face of evidence.
Here's a younger guy saying something similar, entrepreneur and popular YouTube vlogger Hank Green. (Yes, I cite Hank and his brother John a lot, but they often have wise things to say, even though Hank's delivery often makes me think he should switch to decaf.)