Can I power an (extended) LED stripe from both ends simultaneously?
I disagree with tcrosley in the comment above. 

This is a common problem with distributed lighting. My solution is to calculate the maximum current at periodic places within the lighting strip. Use that current value and whatever maximum voltage drop that you can tolerate to determine the maximum resistance allowable between the power supply and the LED strip. 

Remember that you are feeding current both ways on the strip when determining the current. 

Then run two heavy-gauge conductors underneath the LED strip and periodically make connections from the heavy wires to the strip. The wire size is determined by the distance and the maximum allowable voltage drop. 

Note that you can stagger or alternate the connections. That is - connect only the Ground at one location, connect only the power at the next location. This can help make things both easier and neater than trying to make both connections at the same location. 

Feed both ends of the heavy conductors if needed. This can be separate power supplies.