Short answer: yes.
But we need to get a bit technical why.
In modern particle physics, "particles" arise as a result/manifestation of quantum fields. These fields are the most fundamental physical entities we currently understand or claim to exist. The symmetries and interactions of these fields give rise to the particles, forces, and physical nature of our universe.
If there's a deeper or better model/explanation, we don't know it yet or its speculative and unproven.
Why does this matter? Because the atoms, forces, and properties, in our universe come into existence because of these quantum fields. At least one field was deduced to be "maybe existing" to make the rest of them work - and it naturally brings its own particle to the game (the Higgs Particle/Boson).
If another field existed, then it would allow for its own novel particles and interactions to exist. It could radically change behaviour at certain energies ("symmetry breaking", allows stuff like superconductivity to happen, on a macro scale, as well as the visible universe to exist!). We would have new extra Interactions, maybe new extra forces and new ways things can combine or interact (on similar or larger or smaller scales than we currently know of), and existing ones would change. Perhaps subtly, perhaps not. Both are possible.
In fact "dark matter" is widely suspected to be undetectable directly at present, simply because its some novel particle/s, and its laws mean it doesn't interact with most things, so we have very few ways to directly detect it. There could be a whole family of dark matter particles, we just don't know. There could be entire "new" interactions we've never seen, or which only occur visible to us in certain very limited circumstances, and crucially, they could be radically different to those we're used to. I'm going to use that approach, and speculate on a similar mechanism to answer your question.....
Proposed solution
Suppose a new quantum field existed or was undiscovered so far, with some appropriate properties. This is very likely, and not entirely fanciful, but its effects could be.... well, that's up to you to devise. Its properties mean that one or more new particles may also exist. (They might also mean that existing particles can achieve novel states.)
Note that you want it to not fundamentally disrupt existing outcomes, as it takes exceedingly little change to the known laws/interactions, to destroy the fabric on which life as we know it depend. You want it just to add to the possibilities. That's probably possible (a new field wouldn't have to interact in damaging ways, or at all, with known fields) and certainly handwave-able.
If the field allows for new particles, they may not be readily detectable at present. We aren't noticing them. These new particles could have suitable properties so they can combine within the atomic nucleus and create a nucleus with fractional charge, and allow fractional negative charges around it - apparently our belief that charge can only be an integer within a nucleus or in its valence shells is only valid for a Standard Model that excludes this newly discovered field. So you'd have new elements slotted in between, in addition to the current (integer proton count) ones.
Alternatively the field could allow new particles that can combine within an existing ordinary atomic nucleus, without affecting its electrical charge or disrupting the nucleus stability, but now the nucleus has some changed or extra property.
You now have all the physics mechanism needed for your question! Quantum fields can do stuff like that. Handwave some details away, but the hard core should be valid.
