I believe the free version of Ad-Aware has got realtime protection; unless you disable the realtime component in Ad-Aware, there might be conflicts (and they don't have to be necessarily visible or immediately noticeable to you) with Norton. Running two realtime security programs on your machine is not recommended, as they can, for instance, interfere with each other's threat detection mechanisms and actually make your computer more vulnerable to threats that way.
If you want an extra scanning program, you can download the free version, which does not have realtime protection, of Malwarebytes from here:
Yes, it can happen that one program misses some things. That is why we recommend the free version of Malwarebytes and the free version of SuperAntiSpyware. These 2 programs are on demand and won't interfere with your N360.
While one program might catch something the other misses, what usually happens is that both programs try to catch the same thing, or tries to scan the files of the other which causes problems. Neither one will allow that. Sometimes one tries to act against a threat and the other prevents it from working correctly. Sometimes, with new malware, neither program responds. There have been quite a few instances where the second program actually prevented the removal of malware because it did not allow the registry to be changed.
One good antivirus program running in real-time, and one or two on-demand scanners are far better security.
There is no need of Adaware if you have the latest version of Norton as it is having advanced antispyware technology. Adaware also has a realtime protection/scan that runs in the background which may conflict with Norton Antispyware feature. Symantec does not recommend having multiple firewalls/antivirus/Antispyware installed on the same computer at one time. It will be always better if you run only one anti-virus, anti-spyware or firewall program as there is no guarantee for compatibility between different security products. You may encounter unknown problems between the programs and it may also affect system performance. Running only Norton provides the best protection and minimizes conflicts between security programs. Otherwise, you may need to disable the real time protection from one of the security programs and use the other to avoid conflicts in running programs with similar features together. You can have any run-on-demand scanner as a secondary protection if required.