It's not a free pass going to higher pressure.
First off, any given pressure tank has a larger drawdown (tidal volume, amount of useful water stored) at 20/40 than any higher setting. Unless you increase the size of, or add additional pressure tanks, at a higher pressure your pump will stop and start more frequently (which is usually considered to be a large factor in how fast it wears out - running longer and starting less is better than running less each time, but starting more.) As you can see from the example chart below, for a given tank, 40/60 is already less than half the volume stored at 20/40.
This version is not quite as bad (they do vary a bit):
![another tank chart](https://i.stack.imgur.com/meQ0n.png)
You will need to check your tank, but most pressure tanks are good to 100 or 125 PSI (the ones in the upper chart are only 75 though), so that's probably not the limiting factor. You'll need to check what the pressure on your hot water heater's temperature-pressure valve is, but it's probably also fine. My pump supplier suggested (and I installed) a 100PSI overpressure valve on the system as the pipe entered the house.
Many common items (toilet valves, etc) don't really like pressure over 80 PSI (yes, I know you only proposed 70.) 60/80 is about as high as you can expect to adjust most pump control valves to.
It's not the most common situation, but if your well pump is marginal for the depth you may reach a situation where it's cannot reach the higher setpoint as you use water and draw the well level down - so it will sit and spin but not reach the shutoff pressure.
If your system was designed for 20/40, I'd suggest doubling or tripling your tank capacity if going to 40/60, and tripling if going to 50/70. How important that is may be influenced by how deep and inconvenient and expensive your pump will be to change when its time comes. Cheap and easy, maybe you don't bother increasing the tank at all. Expensive and a major hassle - you might consider more tankage a good investment.
Allowing for the possibility that it could be otherwise as diagnosis via internet is imperfect....
I see both a pump problem and a check valve problem here. While they may be located in the same place (if your only check valve is on the pump - which is the way mine is set up) they are not the same thing...though I also see another possibility that would do both.
- Pump problem - pressure not getting above 27 PSI, or 15 with a faucet open.
- Check valve problem - when you shut off the pump the pressure drops
to zero - unless you are using water, the pressure should stay at 27 if it was pumped up to 27. When you only had 10 PSI in the tank, there should have been some water storage - when raised to 38, no water could possibly make it into the tank, since that's designed for 40/60 PSI operation and won't take any water until the the water is over 38 PSI.
The third possibility - you have a leak in the pipe above the pump and check valve - so the pressure is limited, and the water drains away from the leak without the check valve being to blame. If you can go and listen at the top of the well while someone else turns the pump on, you might actually hear this leak, if its in the well above waterline. Or you might have a recently soggy spot in the yard if it's after the pipe leaves the well.
While I don't think it's to blame in this problem, I'd also suggest rechecking your tank pre-charge to see if it has stayed at 38 PSI or started to fall again. Often when the precharge becomes low it's not simply a matter of needing to add air - it's a failure in the bladder or diaphragm.
Best Answer
You can, but it's not all to the good...
Normally, most household plumbing fixtures (especially toilet valves) prefer a pressure of no more than 80 PSI. Your pipes are nearly always not the limiting factor. Most well pump controllers have a fixed 20PSI swing from on to off, but many can be adjusted so they can swing 40-60, 50-70, 60-80, 30-50, 23.5-43.5 etc.
The volume of water stored in your pressure tanks becomes less as the pressure is raised - a captive air tank swinging from 40 to 20 PSI stores far more water than one swinging from 80 to 60 PSI. This matters because more pump starts is one of the factors that leads eventually to pump death and replacement, always a fun time. So if you raise the system pressure, you should probably also increase your storage tank volume to prevent having lots of very short pump cycles. Switching over to a "constant pressure" (or "nearly constant pressure") system is another alternative, though it can be costly.
If you change system pressure, you need to change precharge air pressure in captive air tanks (typically about 2-3 PSI less than the lower cut-in pressure - ie, 58 lb for an 80-60 setting.)
You should observe system pressure as water is used - a properly functioning 40-60 PSI system should drop to about 40, switch on the pump and raise to 60, switch off the pump and drop to 40 as water is used. When no water is used the system pressure should hold steady. If you turn off all other water use and measure exactly how much water is needed to drop from 60 to 40 you can gauge the health and current storage capacity of your pressure tanks. If you measure the time it takes to refill from 40-60 (without drawing any more water) you will know your pumping rate (and if it takes less than a minute, conventional wisdom is that you already need more storage capacity.)