Birth and death records started being kept in Nebraska in 1904. Prior to that, unless there was a local birth,death,marriage ledger kept at the local County Courthouse or other local government office or a delayed birth record was requested and granted, the accepted record prior to 1904 in Nebraska is a birth record kept in a family Bible.
Marriage and divorce records start in 1909. Check with the local County Clerk or Clerk of the local District Court for records prior to that,
Some counties kept records in ledgers before 1904 and 1909. Some of these were lost to natural disasters (tornado, fire, flood etc) over the years, but it doesn't hurt to check with the County Courthouse or local historical or genealogical society, you might be surprised by what is available at the local government level. The Nebraska State Historical Society may have records of interest as well.
From 1904 to 1911 births were typically filed only under the father’s name and the registration doesn't provide the name of the child. These records are not birth certificates and provide only minimal information. Certificates for births started in 1912. Delayed birth records started being allowed in 1941 for those born in Nebraska before 1905.
I'd also recommend checking local newspaper archives and checking with the local library to see what old newspapers they might have available. There are a select few of these old newspapers available online. For Nebraska I recommend the following:
http://nebnewspapers.unl.edu/
Forms to request records from the State Department of Vital Statistics and the Nebraska Statutes that govern such requests can be found here:
http://dhhs.ne.gov/publichealth/pages/vitalrecords.aspx
Most Nebraska county governments have their own websites, with a couple of notable exceptions. You can find contact information for County Clerks and Clerk of the District Court via those individual websites.
