Perturbative series for bosons I have recently read that

... the perturbation series ... is valid only when the perturbed state
  is qualitatively similar to (or ‘has the same symmetry as’) the
  unperturbed state. This means that whenever a system undergoes a change
  of phase — like gas $\rightarrow$ liquid, liquid $\rightarrow$ solid,
  paramagnet $\rightarrow$ ferromagnet, or normal metal $\rightarrow$
  superconductor — since the two states involved are qualitatively
  dissimilar, the perturbation series breaksdown.

[Mattuck, R. D. “A guide to Feynman diagrams in the many-body problem”, 2nd ed. (1992), p. 266]
One can fix this issue by introducing anomalous propagators like in BCS theory. As far as fermions are concerned the presence of Fermi edge ensures that perturbation series converge. 
What about bosons? Are we able to perform perturbative analysis and use diagrammatic expansion, Green function etc. – all these field-theoretical stuff? 
When does the perturbative series method break down?
 A: It's true that "the perturbative series is valid only when the perturbed state is qualitatively similar to the unperturbed state". Generally perturbation theory is acceptable when the coupling is weak, in which case the coupling can be treated as a small perturbation of the free field theory at all energies (for example Yukawa theory and $\phi^{4}$ theory.
The problem with perturbation theory comes when the coupling is not weak, in which case perturbation theory CANNOT be used. This leads to theories with charge fractionalization, confinement, and emergent space (AdS/CFT correspondence).
See Interacting Fields section of Tong Lecture notes on QFT for more information
http://www.damtp.cam.ac.uk/user/tong/qft.html
For the original paper showing that methods of QFT can be used on many boson systems, see Beliaev's paper
http://www.jetp.ac.ru/cgi-bin/dn/e_007_02_0289.pdf
A: 1) I should note that most perturbative expansions that are of interest in physics are not formally convergent (and more often than not, not Borel-resummable either).
2) There are many examples of useful perturbative calculations for bosons. The oldest example (probably) in Many-Body physics is the calculation of the energy per particle of the weakly non-ideal hard-sphere Bose gas, see T. D. Lee and С N. Yang, Phys. Rev., 105:1119 1957. Other examples include the calculation of the critical exponents in $\phi^4$ theory (using the epsilon expansion). In particle physics there are not so many purely bosonic theories. An example would be the calculation of the equation of state of a pure gluon plasma at high temperature (which is perturbative because of asymptotic freedom). 
3) There are indeed some differences between bosons and fermions. At finite temperature, the Bose-Einstein distribution has $T/\omega$ divergence, whereas Fermi-Dirac statistics leads to Pauli-blocking. The $T/\omega$ leads to various IR problems, but in weak coupling these can usually be dealt with using resummation ("screened perturbation theory"). Pauli blocking improves the perturbative expansion in fermionic theories. This is the basis of Landau liquid theory, which suggests that a perturbative expansion may exist even if the underlying coupling is strong. Hoever, i) the Landau liquid parameters are themselves non-perturbative parameters, ii) the Landau liquid ground state is usually ultimately unstable (towards condensation, freezing, magnetism, etc). 
A: 
1) Are we able to perform perturbative analysis and use diagrammatic
  expansion, Green function etc. – all these field-theoretical stuff
  [for bosons]?

In general, the field-theoretic methods (at finite or zero temperature) can be applied to both bosons and fermions with slight differences which originate from the Fermi-Dirac and Bose-Einstein statistics. As an example, you can consult
Fetter, A. L., and J. D. Walecka. “Quantum Theory of Many-Particle Systems” (1971).
The main difference is not in the many-body formalism, but in the physical behaviour of the many-body systems of bosons or fermions; namely, the bosonic ground-state is often (but not always) a BEC while the fermionic ground-state is often a Fermi liquid. The simplest example of a bosonic many-body system appears in the study of quantized lattice vibrations (leading to non-interacting phonons [see e.g., chp. 12 of the reference above]), or the second-quantized form the Maxwell's equations (leading to non-interacting photons).

2) When does the perturbative series method break down?

This is a very broad question. Usually, the perturbative method is valid for a sufficiently small magnitude of the interaction strength.
When the interaction strength is increased, or when new types of interactions are introduced, then the perturbative ground-state can become unstable leading to the formation of a new ground-state with utterly different properties; this is actually a quantum phase transition. For the case of bosons, there could be a transition from a superfluid phase to a Mott-insulating phase (see e.g., Greiner, M. et al., Nature 415, 6867 (2002): 39–44 <PDF>).
In a perturbative analysis, the signature of such phase transitions is usually a divergence in the perturbative resummation in a particular regime of parametres. This implies that the correct perturbative resummation on the two sides of the transition are different; that is, they are performed around different ground-states and include different physical processes. 
For example, in the superfluid-Mott transition mentioned above, 
the Hamiltonian is the bosonic Hubbard model,
$$ H = J \sum_{\langle i , j \rangle} a_i^\dagger a_j + \sum_{i} \epsilon_i \hat{n}_i 
+ U \sum_{i} \hat{n}_i (\hat{n}_i - 1) ~,
$$
where the $i,j$ are lattice-site indices, $a_i^\dagger / a_i$ are creation/annihilation operators for bosons on the lattice ($\hat{n}_i$ is the particle-number operator), $\epsilon_i$ is the local energy of a boson on a lattice site, the term proportional to $J$ is the kinetic (or ‘hopping’) part, and the term proportional to $U$ represents a (spatially-) local interaction between the bosons.
The ground-state is essentially determined by the ratio $U/J$, the relative strength of the local interaction compared to the kinetic energy.
If $U/J \ll 1$, the ground-state is a superfluid, i.e., a collection of itinerant (delocalized) bosons; if $U/J \gg 1$, the ground-state is a Mott insulator with strongly localized bosons. The two ground-states are essentially different, so there should be a transition point at intermediate values, $U/J \approx 1$. The perturbative expansion in the superfluid phase is around the kinetic part, with the assumption that $U/J \ll 1$, while the perturbative expansion in the Mott insulating phase is around the interacting part, assuming $J/U \ll 1$.
This can be understood more clearly by referring to the cited paper by Greiner et al. For a detailed calculation, consult 
Stoof, H. T. C. “Ultracold Quantum Fields” (2014), chp. 16.
With the two sections above, I've tried to concisely explain the quoted paragraph from Mattuck's book.
Finally, I believe that the following statement,

One can fix this issue by introducing anomalous propagators like in
  BCS theory. As far as fermions are concerned the presence of Fermi
  edge ensures that perturbation series converge

is not correct as it stands. First of all, there is no “fixing” particular to the case of bosons; if the ground-state changes, the perturbative expansion should be performed around the new ground-state, regardless of the fermionic or bosonic property of the system, as explained above. Secondly, the existence of a Fermi surface does not guarantee the convergence of a perturbative series for a fermionic system; an example of this would be the Kondo problem, for which perturbation theory fails at low temperatures, but the ground-state is a Fermi liquid with a proper Fermi surface [see e.g., Nozières, P. “The Kondo problem: Fancy mathematical techniques versus simple physical ideas”. Proc. 14th Conf. Low Temp. Phys., part V, Finland (Aug. 1975), 339–374].
