In this scenario, is it feasible to use more than one context?
Yes, both is feasible and recommended for some cases, where the visibility of all entities in a given context is not interesting.
For example, in ASP.NET Identity, a context separate from the rest of the application is usually used.
Can I end up having relationship and data access issues?
It can. For example, if you make two selections in different contexts and relate the entity in some way (as in the case where you want to create associative records from cardinality n to n), there may be inconsistencies because the saved context can understand the object of the other context as being a new object, which does not yet exist, even leading to duplication of data.
Is there a particular reason to use multiple contexts in a single database?
As already mentioned above, in a scenario of authentication and validation of logins is a classic scenario for this separation.
Another reason may be speed. Contexts with many records can degrade application performance. In this case, two instances of the same context are used, one read-only and one persistence, with different properties settings.