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Stimulated by the exciting progress on the observations of the fully-charmedtetraquarks at LHC, we carry out a combined analysis of the mass spectra andfall-apart decays of the $1S$-, $2S$-, and $1P$-wave $cc\bar{c}\bar{c}$ statesin a nonrelativistic quark model (NRQM). It is found that the $X(6600)$structure observed in the di-$J/\psi$ invariant mass spectrum can be explainedby the $1S$-wave state $T_{(4c)0^{++}}(6550)$. This structure may also bearsome feed-down effects from the higher $2S$ and/or $1P$ tetraquark states. The$X(6900)$ structure observed in both the di-$J/\psi$ and $J/\psi \psi(2S)$channels can be naturally explained by the $2S$-wave state$T_{(4c)0^{++}}(6957)$. The small shoulder structure around $6.2-6.4$ GeVobserved at CMS and ATLAS may be due to the feed-down effects from some$1P$-wave states with $C=-1$ and/or some $2S$-wave states with $J^{PC}=0^{++}$.Other decay channels are implied in such a scenario and they can beinvestigated by future experimental analyses. Considering the large discoverypotential at LHC, we also present predictions for the $bb\bar{b}\bar{b}$ stateswhich can be searched for in the future.

Full-heavy tetraquark states and their evidences in the LHCb di-$J/\psi$ spectrum