Can electric /magnetic fields exist without source currents Can electric /magnetic fields  exist without source currents ? If so can a brief example be given? 
 A: Today, we are talking about an EM field, and the electric and magnetic components are observer dependent. This means that the same field could be viewed as electric from one observer's view and as magnetic from another observer's view.
As per our currently accepted theory, the Standard Model, the EM field exist through all of space.
Sometimes we talk about near and far fields. It is important in your question.


In the far-field region, each part of the EM field (electric and magnetic) is "produced by" (or associated with) a change in the other part, and the ratio of electric and magnetic field intensities is simply the wave impedance. However, in the near-field region, the electric and magnetic fields can exist independently of each other, and one type of field can dominate the other. 


Now, if you are looking for the near field in your question, and you are asking whether an electric field can be generated without a current then the answer is yes. But you have to note that as per the comment, the electric field of for example an electron seems static in space.
Still, the electron having rest mass, though it is stationary in space, it is moving in the time dimension. Since it is moving in the time dimension, it could be viewed as a current in time, thus creating an electric field around itself.
Whether this created electric field could be viewed as a magnetic field, is observer dependent. If you have an observer who is moving relative to the electron, this observer could view the field as magnetic.
