In general, mixed breed dogs are good dogs to have. Provided you save a life and get one from a shelter instead of patronizing backyard breeders who give cutesy names to mixes. Adopt one from the shelter and you can call it whatever you want. I had a hound mix that wasn't all too bright. We called her a Hungarian Upid-stay. Amazing the number of people who would say, "Oh, my uncle had one of those."
But before you get any mixed breed dog, best to do some research on the breeds that make up the mix. A border collie/jack russel mix, for example, is going to be a high-energy, hyper dog that needs a lot of physical and mental stimulation. A pug/cocker mix may have breathing problems since pugs have no muzzle to speak of and cockers usually have a shorter muzzle. Also, what kind of grooming would you be looking at? Pugs need their wrinkles cleaned regularly, cockers need routine brushing to prevent mats.
Many people will tell you that mixed breeds have "hybrid vigor" that prevents them from having health problems. A mix generally won't have the same health problems that a purebred would have, but sometimes they pick up the bad genes from BOTH contributing mixes and can end up having more problems. Such as pugs having a tendency towards patella luxation and a cocker having a tendency towards cherry eye. Basically, any dog can have health problems, some are just more prone to them than others.