Try adding -nlt geometry to your ogr script:
ogr2ogr -skipfailures -f "ESRI Shapefile" myshape.shp mygeojson.geojson -nlt geometry
I've experienced the problem you describe using ogr2ogr to translate datasets containing both singlepart and multipart features in the same layer; for example, POINT and MULTIPOINT, LINE AND MULTILINE, and POLYGON and MULTIPOLYGON.
By default, OGR wants you to obey convention and use only one feature-type per dataset. If you break the rule, OGR will throw a generic error and recommend the -skipfailures flag. This can be misleading if the real issue is mixed/matched geometries in the same layer.
Since ESRI shapefile does allow single/multipart features in the same layer, you need to realize OGR may throw this error even if you have valid geometries. In that scenario you have two options: 1) Convert multipart features to singlepart (which ogr can do via the -explodecollections flag), or 2) use the -nlt flag and specify GEOMETRY as the option. The -nlt GEOMETRY option allows you to combine a mixture of feature types in the same layer. Of course, as a consequence, OGR won't won't protect best practices.
If your GeoJSON file "started life" as an ESRI shapefile, it's possible that you have both POINT and MULTIPOINT features contained in that dataset, in which case you might be encountering this issue even with valid points.