Literally the house next door to the first one I purchased is for sale. I was thinking of buying in different neighborhoods though to limit risk by diversifying locations. Thoughts?

It's way easier to manage two close together than two spread apart. If the neighborhood is in good shape, go for it. 

I have a management company though, so that wouldn't make a difference

My husband and I focused on buying neighborhoods at a time.  We were able to maintain the neighborhood kept it's market value.  Plus, as the above poster stated, much easier to keep an eye on your homes when doing open houses, drive by's etc.,  Yes, most definitely advantageous!

I've got two I bought from the builder years ago, they have proven to be problematic times two.  However, I didn't know jack (thought I thought I did) about RE investing when I bought them.   Your implied point about increased risk buying in the same neighborhood is a good one.  

Having said that there are several benefits to having houses close to each other.  Including being able to observe the other house when you or your people are at one of the houses.   You don't have to bone up on and keep informed about two neighborhoods.  If you have an excellent tenant in one of them you can ask them to keep an eye on the other one.  I'd be very judicious about this since it may lead to problems, particularly if you have different ways of treating the tenants.  

You are basically assured any contractor you send to one house will have a coverage area which includes both houses.  

I could go on, but you hit the nail on the head, there are many benefits, but that one risk may outweigh them all.  But you should have a fairly good feel of what is going on with the neighborhood if you have owned the first property for a few years.

How ironic, when I was writing my post I was thinking about nancy's properties in Detroit and what happened to them and many other properties in Detroit.  

This is exactly what I did.  Benefits include being able to check on your property, knowing people in the community that will keep a watch, and you tend to know the area really well therefore easily to find deals.