What happens when an object has the same de Broglie wavelength as the size of an object?

Setting particles aside for a moment, you recall how one probes crystals diffractively, presumably. To probe means to poke around to search into and explore very thoroughly with an instrument of comparable/smaller size: can you probe a pea with a cleaver, or a pomegranate with a tire-iron?

The idea is that the instrument ("probe") you are poking something with has to be quite a bit smaller than the object being probed. In Feynman's analogy, you cannot use a big fat screwdriver bigger than a watch to probe the innards of it: you need a very delicately small one, or, lacking that, you smash two watches together and monitor the cogwheel ejecta.

The Compton wavelength of the probe particle goes inversely as the energy, so it must be quite smaller than the size of the nucleus (fermis, or femtometers). Can you estimate the energies required? You can probe non-destructively, as in the crystallography gambit above, or destructively, by smashing up the nucleus, like smashing two watches with each other.

This is the central theme in subatomic physics: Higher energies and momenta, probe (explore) smaller distances, a seat-of-the-pants appreciation of Fourier analysis and the uncertainty principle.