Suspension of Disbelief

Any fiction series, but especially any Science Fiction (like Star Trek) or fantasy series, require some willing suspension of disbelief from it’s readers, viewers, etc. That is, some things that look impossible, and are barely explained, but we have to assume they happen in the universe. We accept that to follow the series.
In Star Trek, we usually have to accept some science principles that seem impossible, or engineering apparels that defy physics. Some of them, however, look plausible. Ok, we don’t have the technology. But it’s several centuries in the future, they might have invented it. We can even accept the Technobabble (we already talked about it here) that leads it to them.
But some of them simply don’t. From my point of view, we are able to accept any set of hypothesis, if they support an interesting story. But what we are not accepting is inconsistencies. As long as everything is consistent, we are happy. But if we either have some of these hypothesis contradicting themselves, or something that we don’t understand but it’s not explained, we disprove the way they treated that.
What do you think about that? What are the thinks that make it harder for you to keep your suspension of disbelief? Those moments in Star Trek that seem impossible, even accepting it’s the future and they might have invented several new technologies, or discovered new principles.
Tell us here! And maybe it will be the topic of a future “Ask the Science Department” article.