The real check you want to do is just this. Does the system run on the battery that it doesn't boot on. Obviously you can't boot from the battery but if it runs from that same battery then its a motherboard problem. Also, note that your machine shouldn't be able to resume from standby on battery either. This problem has happened to many other Dell users including myself (twice). The new motherboard completely fixes this issue.
Note: Your machine is probably still able to boot from battery. It does so very infrequently; about once in every 20-50 tries. Obviously, this is still considered unusable to most people.