Can you get pregnant 3 days after getting a Depo shot?

Oh my goodness. 3rd time of using emergency hormonal contraception in a month. Your poor body.
Your emotions are likely to be all over the place, & you could get some intense hormonal side effects for sevral months. Anything you might get with a period or pregnancy, or other hormonal change in your body, but likely more intense.
It's meant in total emergencies, & certainly not more than once per menstrual cycle. There are no studies done on the long term effects of such frequent use. I.e. no one knows what effects it may have on your body in years to come. It's also not as effective as regular hormonal birth control. It's simply worth a try in an emergency.

With your first Depo Provera shot, if you get it during the first 5 days of your period, you're protected against pregnancy immediately. Otherwise, it's 7 days before you're protected.

Do an Internet search on Correct Condom Use. There are ways to stop them splitting.

You'll need to do a pregnancy test in a few weeks, to ensure you aren't pregnant, as it's thought that Depo Provera can cause harm to an unborn baby. Mixed medical opinion.
If you are pregnant, you'll need to see your doctor immediately.

There's quite a bit to know about Depo, & I don't know what you know already, so I'll give you the lot & you can take what you need from it...
My advice is to cut & paste this & keep it somewhere safe, so that if you have any questions, you've got it handy.

Depo Provera works by convincing your body it's pregnant, even to the point that you can get pregnancy symptoms/side effects. Lower back ache is common, as your pelvis ligaments soften, ready for a growing baby. Leaking breasts can also happen from time to time.

It's common to get irregular bleeding for the first 2 to 3 shots, & then you typically find that your periods stop, altogether, until you come off Depo Provera. Sometimes you can get a little irregular bleeding in between shots, especially after sex or around the time that my shot was coming up to be due.
However, because you've taken so much emergency contraception, there's no knowing how you'll be effected. It may be longer before the side effects from Depo affect you, or it may be sooner, & more intense. You may have no bleeding for months, & then get some. Anything goes, really. No one can tell you.

The Manufacturer advises that you should not be on Depo Provera for longer than 2 years, unless you have no other option for contraception. This is due to the significant risks that come with the Drug. In particular, Depo Provera is known to cause bone thinning, leading to osteoporosis. There are other conditions associated with Depo, such as certain cancers, for example, however, there are such risks with all hormonal contraceptives, so it's just a case of making an informed decision.

The Manufacturer also states that if you have to stay on Depo Provera for the long-term, that you take regular breaks, every couple of years, to give your body a rest, & that you have an MRI scan after a significant time on the Drug, to monitor your bones. I've just been through this, & have to have the scan repeated when I reach menopause, which is when any bone damage is most likely to show-up.
I wasn't told about the breaks every couple of years, but luckily I did take some breaks, & always took a good calcium, magnesium & vitamin D supplement, whilst I was using Depo, so I'm hoping that this will prevent any bone damage. it's necessary to take the vitamin D & magnesium, in order to properly absorb the calcium.

The Manufacturer also advises that even after just 1 shot, you may be infertile for up to 18 months. Don't rely on this for contraception timing, & still get your shots every 12 to 13 weeks (12 is best as it gives you a week's grace, should you be unable to get your shot), but you may find that you either don't get periods for quite some time after stopping Depo Provera, or that they are very irregular. Sometimes, you can even get very heavy & constant bleeding, after Depo Provera. I had that, too. We hear of many women who's periods struggle to get back to regular cycles, at all, after Depo Provera. I managed to sort mine out, after 2 years of medical procedures & intervention, by using Vitex. Worth writing down, in case you need it in future. You can get it at health stores or online. If I'd known about it earlier, I woudln't have had to go through all the medical intervention.

Most of the information I'm giving you can be found on the sheet that comes with the shot, & also on the Manufacturer's website. In my experience, the doctors/nurses rarely give you this sheet. You can ask for it, though. Otherwise, just go to the Manufacturer's website (Pfizer), & look for the Depo Provera page, & the sheet to download, called 'Depo Provera CI U.S. Physician Prescribing Information'. I'll try to add the link in the comments. Sometimes they'll let me, sometimes they won't.

Common side effects are basically anything you might get with a period or pregnancy, or other hormonal change in the body. Weight gain is one of the most commonly reported, as is the lower back ache, but I also got less common side effects, too, such as very painful intercourse, & constant cystitis, which both stopped after a year or two of coming off Depo Provera. It's a shame it took 18 years of suffering, to find that out...