If it sounds like I am doing a series on the concept of "brick walls" you are right. I have a whole series of classes this month at the Mesa FamilySearch Library on the subject. I would guess that few of the people who come to me with their genealogical problem cases have gone through a research checklist of questions to ask before you give up or claim a brick wall. Here are some of the questions I would ask.
Have I carefully checked the location where your ancestor was supposed to live?
Most of the claimed brick wall issues turn out to be situations where the researchers are looking in the wrong place or have yet to identify the place. Whenever someone comes to me with a question, I immediately examine where and when the event they are looking for supposedly occurred. It is no longer surprising to me to find out that they have been looking in the wrong place all along. Usually, it is a jurisdictional issue, with the county or the city being different than the one they have identified in their passed down records. I use the Newberry Atlas of Historical County Boundaries and other mapping programs, to start to verify the county boundaries at the time of the supposed event.
What do I know about the specific ancestor's family, especially children and grandchildren?
Sometimes people claim that they have hit a brick wall when all they really have hit is one brick. Constant attention directed at the missing person is unproductive. You already know that person is hard to find, so try someone else; brothers, sisters, neighbors, children, grandchildren. You need to learn as much as you can about the entire family and any records that may have survived from family or neighbors.
Have I looked in the newspapers?
I keep asking this question and the answer is almost always no. The number of online digital newspaper collections increases daily. You can also see which newspapers were published in the state and county by going to Historical American Newspaper on the Library of Congress website. There are also several very large online subscription websites.
Have I read a good book lately?
What I mean by a good book is one that talks about the history of the location where my target ancestor lived. Check your local library or historical society for suggestions or go online and search in WorldCat.org for the location plus the words genealogy or history. You might find your ancestor's family in the book or not, but you will come away with a greater appreciation for the area where they lived and just might get some good ideas about where to look.
Have I followed the suggestions in a record selection table?
If you haven't looked at a record selection table, especially for the country or state (province, district etc.) where your ancestors lived, you probably have no idea of what is available. I suggest the United States Record Selection Table on the FamilySearch.org Research Wiki as a starting place.
I am sure I will have further suggestions as the month progresses.