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At 4dpo, no. At 7dpo, yes although rare. Although implantation can happen anytime between 6dpo and 12dpo, it usually takes place around 9-10dpo, and then it can take a few days for hcg levels to be high enough to give you a positive hpt.

Ok thank you. I think I may have my ovulation off by a few days because I'm already feeling nauseous and slight cramping and so very tired. Does that sound promising?

At 4dpo. No it is not possible.
The absolute easiest implantation can occur is 5dpo (though this is rare!) this is because the egg must 'hatch' from the protective layer around it (the zona pellucida) before implantation and this does not happen until 5dpo.
Even after that, a recent study showed that the earliest implantation was 6dpo and then only 1% implanted that early.
After implantation, it's another 2-3 days before the HCG would build up enough to be detected! So a BFP at 7dpo is possible, but rare. As studies show days 8, 9 and 10 dpo are the most coming days for implantation to occur.
Good luck with testing :)

I had a positive at 4 dpo back in November 2014. Unfotunately it was a chemical. So yes its possible. But your ov date could be off

My theory is that I implanted really early, causing a positive test really early, but the early implanting also caused the chemical

If you were to implant at 4dpo, which shouldnt be able to happen (again because of the zona pellucida) then you would likely be implanting in the Fallopian tube, resulting in an ectopic pregnancy. As it takes approx 5+ days for the fertilised egg to travel through the Fallopian tube to the uterus.
If you had a positive at 4dpo is was either a dodgy test or your dates were out.