There are definitely some other disorders you can get as a consequence of having OCD, such as Depression, Panic Disorder, Social Anxiety, and suicidality. However, schizophrenia and psychosis is not one of them.
But many OCD’ers worry that they may get schizophrenia or that they are in the process of getting it. Sufferers call this S-OCD, Schiz OCD or better yet OCD with the Schizophrenia or Psychosis Theme. This is simply someone with OCD who has adopted the theme of a fear of going psychotic. As with most other forms of OCD, the fear can cause symptoms that seem to mimic the fear itself. In this case, it can cause symptoms that mimic schizophrenia or other psychoses on the surface, however, careful prodding and questioning generally makes a differential diagnosis between OCD and Schizophrenia fairly straightforward.
Nevertheless, many S-OCD’ers sadly get diagnosed with schizophrenia or other psychoses by ignorant clinicians and as a result are medicated inappropriately. This subtype of OCD is very poorly known and often misdiagnosed.
I run into S-OCD’ers with incorrect diagnoses of Psychotic Depression, Schizoaffective Disorder, Schizophrenia, etc. on a fairly regular basis. The fact that when OCD is very bad, OCD’ers appear psychotic on the surface (but are not psychotic) confuses matters even more. It takes an experienced clinician to figure out what is OCD appearing psychotic and what is an actual psychosis.
At times the two illnesses are found in the same person, and sometimes in these cases it can be hard to figure out where the OCD ends and the schizophrenia begins or figuring out if a given symptom is best seen as one illness or the other. When the illnesses occur in the same person, it is sometimes called schizo-obsessive disorder. These people, who have much better insight than other schizophrenics, sometimes have a tendency to hide symptoms, which makes diagnosis even more confusing.
But having OCD is not going to give you schizophrenia. That’s not possible.