I've labeled your photo, which might help you understand what's going on.
The Panel Rating
The panel is rated to support 125 amperes, when connected to a 120/240 volt 3 wire system. This means that 125 amperes can flow through each of the upper bus bars and each main lug, without anything melting or catching fire.
The Upper Section
The top two double pole breakers are before the "main" disconnect, which means they will always have power when the wires feeding the panel have power. I'm guessing there is a disconnect ahead of this panel, maybe at the meter or as a standalone disconnect. Based on their size, I'm guessing one (50A) is for an electric stove, electric heater, or maybe a subpanel. The other (30A) is likely for a dryer, water heater, subpanel, or some other appliance.
The Lower Section
The next double pole breaker (50A) controls the flow of electricity to the lower section of the panel. Unlike the double pole breakers above, this one should not have any terminals where wires can connect. The lower section is rated for a maximum of 100 amperes, so the breaker protecting it must be 100 amp or less (50A in your case).
Branch Circuit Section
The lower section (highlighted in purple) is where the branch circuit breakers connect, and has a maximum breaker size of 70 amperes. This means the largest breaker that can be connected in this section, is a 70A breaker.
Upgrade Required?
Since there is only a single available slot (3), it's not likely a central A/C system could be connected without moving things around (at the very least). Upgrading the panel might be your only option, but it would depend on the existence (or lack thereof) of a sub panel, and the existing service provided to the building (and availability of services in the area).
A subpanel may be an option, however, it would require more information than you have provided here.
Depending on the service provided to the building, upgrading the panel may include an upgrade to the service. Installing a new 125A panel is useless, if you don't also have the service upgraded to support 125 amperes.
To determine if an upgrade is required, contact a local licensed Electrician to do a load calculation on the building.
Breakers Protect Downstream Wiring
Short-circuit and/or overcurrent protection devices (breakers, fuses, etc.), are designed to protect the wiring downstream (after them in the circuit). For example. If your panel did have a main breaker, it would be sized to protect the panel wiring. The breaker would not be sized to protect the wiring feeding the panel or anything before the breaker, only the wiring after the breaker.
Summing Up Breaker Ratings Means Nothing
If you total up the rating of the breakers in the branch circuit sub-section, you'll find that you have 85A on leg A, and 105A on leg B. Which means... Well, absolutely nothing. The only limitations here are that there can only be as many breakers as will physically fit, and no one breaker can be larger than 70A.
If more than 15 amperes flow through the breaker in slot 4 , that breaker will trip (open). This protects the wiring connected to that breaker. If more than 50 amperes flow through any combination of breakers on either leg A or B in the branch circuit sub-section, the sub- section breaker will trip (open). This protects the wiring between the sub-section breaker, and the branch circuit breakers.
You could theoretically. have breakers totaling 1,000,000A. It still wouldn't matter, as long as you have proper overcurrent protection.
If I've missed anything, or haven't explained something properly. Feel free to ask additional questions, or point out mistakes in the comments below.
Best Answer
Loose connections can cause a lot of problems, including:
So if tightening the connections solved the problem then you should be all set.
In addition, as you suspected, simply replacing a breaker with a larger breaker is not to be done on a whim. It should only be done when:
With a 20-year track record plus tightening connections solving the problem, all indications are that upsizing the breaker would be a very bad thing to do.
The appliance guy blaming the breaker (or more likely "appliance is OK so this must be something in the electrical system") is actually reasonable. An electrician looking at it and recommending replacing a breaker with a larger one based on an overcurrent trip after 20 years is being irresponsible.
Anecdotally, I suspect the appliance guy and the electrician are both getting relatively underpaid by the warranty company - they (generically, not one particular company) are known (at least in my area) to be a bad deal all around - and therefore likely to end up with the "low bidders". Who may not be quite as qualified as you would hope. Just saying...