Some organizations have faced serious obstacles for disseminating their data according to the Open Data requirements and characteristics, e.g., for everybody and any use. Often this is the case when the data is of low quality, has (potentially) sensitive information, or has non-interoperable data format and semantics. Not being able to (completely) satisfy the Open Data requirements may have made such organizations to appear incompliant with the ideals and objectives of Open Data, despite their full commitments and efforts for data opening. In this contribution we propose a new paradigm -- called Semi-Open Data paradigm -- in order to frame, acknowledge, and encourage such initiatives and efforts that strive along the Open Data objectives but do not comply with Open Data requirements completely due to some practical constraints. For the proposed Semi-Open Data paradigm we further present an assessment method to measure and categorize Semi-Open Data initiatives objectively. This method offers a better way to assess and reward the extent of organizations' efforts to meet the Open Data characteristics than the current method that checks whether all Open Data requirements are met or not (i.e., by making a binary decision).