How long an oil lasts varies from engine to engine. Some are operated in worse conditions than others. Idle and running time, ambient temperature and humidity, stress (load/RPM, etc.), quality of gasoline used, efficiency of air and oil filters, mechanical condition of engine and others are all variables that determine how quickly the oil should be replaced. The 3,000 mile thing is an age-old and obsolete rule of thumb formed way back when oils (particularly conventional) were nowhere near as advanced and robust as they are now. The worst conventional oil today is worlds better than the best of those days. The bottom line is you can't change your oil too often as long as you can afford it. The 3,000-mile rule remains as a safe minimal generalization for all engines.
My best advice when running synthetic oil is to watch it. Since synthetic oil doesn't break down, it mainly needs to be changed to get the contaminants out of the engine. Yes, it loses some components over time but that's a whole other subject, thread, forum. Watch the oil, and when it gets dark, it's dirty and should be changed. If your engine is operated in optimal conditions, you should be able to get at least 5,000 miles out of it. If sticking to a 3,000 mile OCI keeps you sane and going longer would worry you, then keep with it.
I believe you would be perfectly fine running a cheaper synthetic or even a quality conventional oil and changing it at 3,000 miles or even 2,000 if that comforts you. Personally, I'd run a top-quality synthetic and longer OCI (~5,000 miles) because it has been proven time and time again that synthetic oil protects better and for longer than conventional oil. Running a quality filter (I prefer Wix) and a 5,000-mile OCI satisfies me financially and, judging by the oil's appearance, isn't too soon (wasting) but isn't too long (it's not horribly dirty).