Can I connect power supplies to each end of an LED strip?
Basics

A 12-volt, 300-LED 5050 strip needs 6 amps, or 2 amps per RGB channel.   That's 72W. 

DC power supplies can be ganged in parallel if they're  of any quality. 

On an interconnected system, there must be only one controller, period.  

A system can be of any size, if extended by amplifiers and/or feeders.  So if you want to put mood lighting on the USS Intrepid, good news.  

Amplifiers take the output of the one controller and use its signal to drive more strips.  This is used if the strip load exceeds the controller's capacity, or if you want/need to use multiple power supplies.

Everything I said above applies to the eBay cheapies. You don't need to go to exotica for any of it.  

That kit is random leftovers

The stuff isn't even meant to work together. Obviously they're supplying 3A power supplies with 6A LED strips. That's why it's dimming out on you.   If you ganged both dc power supplies, you might have enough power to operate one 5m spool properly. 

You assumed  it would work together merely because it was sold together.  Nope, those people do not care.  You have to watch your wallet when dealing with that bunch.   

That said, you did not get ripped off.  The parts you can use, 2 reels and 1 controller, are worth the price on your link. 

What you really need

Your 45’ loop will take almost all of 3 whole 5m/16' strips, or 18A, or 216W.  Per RGB channel, 6A or 72W per channel.

You will not find a controller that can drive 216W.   However, you can have the controller drive amplifier(s).  You can obtain a single amplifier that can drive 216W.  Or you can segment the strip and drive it with many smaller amplifiers, driven by multiple power supplies (or one), which may be easier to hook up. 

You cannot drive 3 strips from one end.  The internal "wires" (PC traces) inside the strip cannot carry enough current to power more than barely 2 strips (10m) and even then, they will "dim" dreadfully down their length. 

If you are seeing dimming down even one 5m strip, I would not be surprised.  Passing 6A (round trip) is a lot for those narrow PC traces. In that case, either split the strip or feed it from both ends.

Option 1: big power supply, 1 amplifier, feeder

Here you use at least a 300W DC power supply and the controller feeds one big amplifier.   The internal PC board traces on the strips can't handle nearly that much power, so you run parallel feeder wire.  This is 4 wires (RGB and common) that run parallel to the LED strips and tie back into it wherever feasible.  At the least I'd feed each end of each strip.  

Option 2: split the strips and use smaller amplifiers

I notice they make some very inexpensive amplifiers with 4A rating.  That can comfortably power half a 5m strip, so you would cut the strips.   You would need to feed each amplifier from the one controller.  Each amplifier could be powered from its own power supply, or from the same supply, or any mix you please. For instance you could use two of those 3A supplies you have to power two half-strips, and then power the rest from a single larger supply.