You have either created a new part to insert onto a page or you insert an existing part onto the page. After click Select to Insert or Save on the new part, the part is not successfully added to the page. Instead, the part seemingly disappears and is not visible. You are able to search for the part by clicking in the content pane and selecting Insert Part or by going to Site Explorer>Parts, but the part will not insert onto the page.
This can happen if the part is being inserted onto a deleted page. In order to see parts added to page, the page must not be deleted.