Well, if you don't have enough data, but I guess you already knew that. Maybe if you clarify your question, I could help. I must admit I don't really understand what answer you are looking for here.
That would depend on the question. While researchers have always tried to find ways to develop new scales to measure constructs, there are always certain constructs or problems that (at least as of now) do not have scales to measure or quantify them. This is why qualitative research exists. However, as researchers, one must always try to develop scales to fill this gap. I recently developed a scale to measure social media jealousy that is free to use- https://www.qeios.com/read/SWMC6H.2
Yes, it’s possible to encounter problems where no suitable statistical method exists, particularly when data is unavailable, too noisy, or doesn't meet the assumptions required for standard techniques. In such cases, alternative approaches like qualitative analysis or domain-specific methods may be necessary.