All residential panels are like that. You'll never max out all the branch circuits at once (which would be 350 Amps), so it's OK to have them total to more than the total panel load.
Think about what each breaker is protecting. The 150 Amp breaker is protecting the wire from the meter to the panel, which can handle no more than 150 Amps, total. The branch circuit breakers are protecting the branch wiring, each one of which can handle 15 Amps, or 20 Amps.
Where you see two breakers bridged like that, it's for 240 Volt service (vs. 120). You've got those for very high power appliances, like air conditioners, electric dryers, electric ranges, and electric water heaters.
You can Google "split phase power" for an explanation as to how they provide both 120 Volts and 240 Volts from the same panel.
The only way that there would be 350 amps would be if everything attached to the system was on at once. This is never the case. As for the breakers you describe they are for 240 volt circuits. 120 volt circuits only require a single breaker.
After reading all the answers this is what is happening. Most are correct in that they say you will never use all circuits at the same time. Each circuit is rated individually at 15, 20, 30, 40, 50 amp service. The higher the rating the more amperage it can withstand. A 15A circuit usually will trip if the sustained amperage exceeds 15A. Small bursts and surges should be able to be absorbed by the breaker. The same thing extends to all circuits including the 150A service breaker. Dual breakers, ie 20/20 and 30/30, are used for 220 loads like HVAC, dryers and electric ranges. The amperage of the breakers state the safe working range of that circuit. Again, small surges and bursts should be absorbed by the breaker, but sustained surges or shorts would trip the breaker.
The fact that you added them all up was nice but means nothing.
It's called Diversity Factor and it happens all over. Each tripswitch is good for its max load. In the unlikely event of even half the circuits being fully loaded, the main breaker would trip so it's safe to use. If you wanted to add lots of extra circuits, you'd be advised to get a bigger panel rated for a higher current. It doesn't make sense to add all the breaker values because they will never be fully loaded all at the same time.
That's not the way it works. Each breaker has a number that tells you what amperage it takes to trip it. It's a safety device that will trip at close to that amperage to keep from burning something out or causing something to overheat and catch fire. The 150 amp breaker is to allow you to turn off the electricity to the entire house and also to trip in case something happens to the panel of breakers such as a short within the box itself that could be caused by tornado damage or high wind damage. The next safety breaker is at the electric companies service pole usually on their own right of way.
You won't use all circuits at full capacity all the time.
That's how you add up to more than 150
how to add up breakers and panel rating ==here is some help for you ... and everything is not on same time ..... (1) No. The sum of the breakers is meaningless. And even if it did mean something, left and right sides are meaningless (they do not correspond to the legs of your power).
(2) If you have no idea what might be plugged into an outlet (i.e., it's just for "general" use), the a standard rule of thumb is no more than 10 outlets on a 20-amp circuit, or 8 on a 15-amp circuit. Count each lighting fixture against these numbers too (although it is usually better to keep lighting and receptacles on separate circuits). Lean to the conservative side if you have the panel space to do so.
(3) You may choose one and only one of the following options: (a) put everything (lights, fan, receptacle) in one bathroom on one 20-amp circuit with nothing outside that one bathroom on that circuit, or (b) put multiple bathroom receptacles on one 20-amp circuit, but nothing except bathroom receptacles on that circuit. Option (a) is usually easier.
(4) Maybe, maybe not. Some fixtures have built-in compartments for the connections, in which case no box is required. How to weatherproof and how much to weatherproof depends on how exposed the fixtures are.
The NEC (National Electrical Code) recognizes that not everything will be used at once, 150A is the total capacity calculated that likely would be used at any time. The wire feeding the panel and the thickness of the metal conducting components in the panel are based on the size of that breaker. The code makes an estimate of the highest consumption likely to be needed based on square feet, non-coincidental loads and various percentages of specific equipment installed. Actually the number calculated is usually much higher than reality, I can't tell you the number of times I have put an amp clamp on service wires of a 200A service and found it only using a 25 amps.
But fyi, Your service consists of two opposing 120 legs (kind of like +120v and -120) that add up to a total of 240v. 150A is available on each of the opposing 120v legs. "A" and "B" legs alternate down the panel from top to bottom, the double handles are then connected to both legs, supplying 240v of force to that connection. If trying to pointlessly add the connected loads then you would have to add the two separate legs separately by adding every-other breaker counting the doubles as two separate singles.
The breakers are individual circuits, you don't add their amperages together.