A core-collapse ( CC ) supernova ( SN ) of Type IIn is dominated by the interaction of SN ejecta with the circumstellar medium ( CSM ) . Some SNe IIn ( e.g . SN 2006jd ) have episodes of re-brightening ( ” bumps ” ) in their light curves . We present iPTF13z , a Type IIn SN discovered on 2013 February 1 by the intermediate Palomar Transient Factory ( iPTF ) . This SN showed at least five bumps in its declining light curve between 130 and 750 days after discovery . We analyse this peculiar behaviour and try to infer the properties of the CSM , of the SN explosion , and the nature of the progenitor star . We obtained multi-band optical photometry for over 1000 days after discovery with the P48 and P60 telescopes at Palomar Observatory . We obtained low-resolution optical spectra during the same period . We did an archival search for progenitor outbursts . We analyse the photometry and the spectra , and compare iPTF13z to other SNe IIn . In particular we derive absolute magnitudes , colours , a pseudo-bolometric light curve , and the velocities of the different components of the spectral lines . A simple analytical model is used to estimate the properties of the CSM . iPTF13z had a light curve peaking at M _ { r } \lesssim - 18.3 mag . The five bumps during its decline phase had amplitudes ranging from 0.4 to 0.9 mag and durations between 20 and 120 days . The most prominent bumps appeared in all the different optical bands , when covered . The spectra of this SN showed typical SN IIn characteristics , with emission lines of H \alpha ( with broad component FWHM \sim 10 ^ { 3 } -10 ^ { 4 } \leavevmode \nobreak { km \leavevmode \nobreak s ^ { -1 } } and narrow component FWHM { \sim 10 ^ { 2 } \leavevmode \nobreak km \leavevmode \nobreak s ^ { -1 } } ) and He I , but also with Fe II , Ca II , Na i D and H \beta P Cygni profiles ( with velocities of \sim 10 ^ { 3 } { km \leavevmode \nobreak s ^ { -1 } } ) . A pre-explosion outburst was identified lasting \gtrsim 50 days , with M _ { r } \approx - 15 mag around 210 days before discovery . Large , variable progenitor mass-loss rates ( \gtrsim 0.01 \leavevmode \nobreak M _ { \sun } \leavevmode \nobreak yr ^ { -1 } ) and CSM densities ( \gtrsim 10 ^ { -16 } g cm ^ { -3 } ) are derived . The SN was hosted by a metal-poor dwarf galaxy at redshift z = 0.0328 . We suggest that the light curve bumps of iPTF13z arose from SN ejecta interacting with denser regions in the CSM , possibly produced by the eruptions of a luminous blue variable progenitor star .