I haven't seen Altium 2003 since 2004 so my reply will be general. If you have two nets on the schematic with different names that are connected together when it is converted to a PCB layout Altium will assign both nets one of the two names. For example you have VCC1 and VCC2 on the schematic and they are connected together when you go to PCB, both nets will be named VCC1 or VCC2 and the nets combined into 1 net.
As a diagnostic; in the schematic assign names to both nets in question, and update the PCB layout to see which net survives.
In the recent versions of Altium in the PCB editor under "Design > Netlist > Configure Physical Net" it would show conflicts and allow you to decide how they are handled. I do not know if Altium had this feature 11 years ago.
BTW - there are times when you may want to "connect" two nets on the PCB. An example is where GND1 and GND2 come together at a common point in a Star grounding scheme. In this case you create a symbol with 2 pins and a foot print with the two pads connected together. This is referred to as a net connector.