New tetranuclear and octanuclear mixed-valent cobalt(II/III) pivalate clusters, namely, [NaCo4(O2CCMe3)6(HO2CCMe3)2(teaH)2(N3)]·2H2O (in two polymorphic modifications, 1 and 1a) and [Co8(O2CCMe3)10(teaH)4(N3)](Me3CCO2)·MeCN·H2O (2) have been synthesized by ultrasonic treatment of a dinuclear cobalt(II) pivalate precursor with sodium azide and triethanolamine (teaH3) ligand in acetonitrile. The use of Dy(NO3)3·6H2O in a similar reaction led to the precipitation of a tetranuclear [NaCo4(O2CCMe3)4(teaH)2(N3)(NO3)2(H2O)2]·H2O (3) cluster and a heterometallic hexanuclear [Co3Dy3(OH)4(O2CCMe3)6(teaH)3(H2O)3](NO3)2·H2O (4) cluster. Single-crystal X-ray analysis showed that 1 (1a) and 3 consist of a tetranuclear pivalate/teaH3 mixed-ligand cluster [CoII2CoIII2(O2CCMe3)4(teaH)2(N3)]+ decorated with sodium pivalates [Na(O2CCMe3)2(HO2CCMe3)2]- (1 or 1a) or sodium nitrates [Na(NO3)2]- (3) to form a square-pyramidal assembly. In 2, the cationic [Co8(O2CCMe3)10(teaH)4(N3)]+ cluster comprises a mixed-valent {CoII4CoIII4} core encapsulated by an azide, 4 teaH2- alcoholamine ligands, and 10 bridging pivalates. Remarkably, in this core, the μ4-N3- ligand joins all four CoII atoms. The heterometallic hexanuclear compound 4 consists of a cationic [CoIII3DyIII3(OH)4(O2CCMe3)6(teaH)3(H2O)3]2+ cluster, two NO3- anions, and a crystallization water molecule. The arrangement of metal atoms in 4 can be approximated as the assembly of a smaller equilateral triangle defined by three Dy sites with a Dy···Dy distance of 3.9 Å and a larger triangle formed by Co sites [Co···Co, 6.1-6.2 Å]. The interpretation of the magnetic properties of clusters 2-4 was performed in the framework of theoretical models, taking into account the structural peculiarities of clusters and their energy spectra. The behavior of clusters 2 and 3 containing CoII ions with orbitally nondegenerate ground states is determined by the zero-field splitting of these states and Heisenberg exchange interaction between the ions. To get a good understanding of the observed magnetic behavior of cluster 4, we take into consideration the crystal fields acting on the DyIII ions, the ferromagnetic coupling of neighboring DyIII ions, and the intercluster antiferromagnetic exchange. For all examined clusters, the developed models describe well the observed temperature dependence of the magnetic susceptibility and the field dependence of magnetization. The computational results apparently show that in cluster 4 two DyIII ions with similar nearest surroundings demonstrate single-molecule-magnet (SMM) behavior, while the strong rhombicity of the ligand surrounding hinders the SMM behavior of the third DyIII ion.