Depends a lot on what sector of psychology you are going into and also the setting in which you would like to work. For many specialty tracks (e.g. neuro, forensic, child/family etc.) many post-docs will have a requirement that their applicants have completed an APA-accredited internship. Also licensing boards in certain states and board certification standards require completion of an APA-accredited internship (or you have to prove that you received the equivalent of that under a qualified supervisor). I think general and school psychology tracks are more lenient when it comes to these requirements. Also, it will be easier to get by without an APA-accredited internship if you are planning on going into private practice versus if you are trying to get a position in the VA, an academic medical center, or top tier hospital etc.