I had new landscape lighting installed with 16 lights totaling 280W on a
300W transformer. The last 7 lights on the run are are ON but "dim". I think there are two different gauge wire connected. Is this a problem or could there be a improper connection?

Sounds like Low voltage lighting.
12 Volts drops fast.
I have had to use number #6 before around the outside of a very large screened in pool enclosure, then tap off with #12 and go up to each light.
What are the distances?
If you didn't want to change the wire, may use lower wattage bulbs at the start?

Strat is on the right track, length of the run really plays in here. In most applications with long runs many manufacturers recommend not making a single run of the wire but actaully making a complete loop so both ends of the wire are connected to the transformer. Extra expense for double the wire but that's what it takes to get the lighting you want.