I'm a big fan of the online sites with digital images of newspapers, but one has to use them with care and forethought and not blindly enter in names and hope. Sound genealogy methodology requires that the researcher know what they are searching as they search it.
Always look to see what newspapers the sites actually contain and what years and specific issues of those newspapers they contain. It is not uncommon for sites to have a smattering of issues within a stated time frame (ie. 1860-1920) that implies an extensive coverage that may in reality be more limited.
In this post we will look at three of the fee-based sites. These are all sites to which I have a subscription and which I use regularly.
Newspapers.com has a wonderful collection of digital newspapers, but be aware of what it has and what it does not. The screen shot below indicates the years for which issues of the Galesburg Register-Mail are available on the site. I know there are microfilmed copies of newspapers in the 1940s because I have used them. This explains why I was not successful in locating an 1947 obituary for a family member.
Even within the issues for a certain year there may be issues that are not digitized as this illustration on Newspapers.com for 1960 shows.
There are a smattering of Davenport Daily Leader issues for March 1904 available digitally on Ancestry.com. I suspect that these issues are available on microfilm as I've used Davenport newspapers extensively on film at the Davenport Public Library.
This screen shot from GenealogyBank shows what issues are available on their site for Evanston, Illinois. I did not take a look to see what individual issues were available.
The free sites may also have similar issues as well. The researcher does not know until she looks at the inventory. How complete the coverage is is something that the researcher needs to know.
And if the site doesn't have a listing of what newspapers are in their database--ask.
Not knowing, not paying attention, or not caring to find out may be why you cannot find the item of interest.
It is very possible that issues in the "gaps" may actually be on microfilm. Don't assume because the online site has gaps in their coverage that the newspapers are not available.
It is appreciated when the the site makes their inventory available to the user.
Local libraries, historical societies, or genealogical societies may be able to assist you in determining where and how to access all extant issues of newspapers in your area of interest.