OP, icecreamguy posted a link to another thread where there is a fair amount of troubleshooting explanation there. It will help you.
Though if you want my best-guess then I would put $10 on the ADSL dropping during the download and having issues retraining to the DSL. Often, ISPs want to "show off" and give their customers the absolute fastest speed internet that they can at the cost of stability and reliability. My bet is that, during the download which slams the line, the modem is losing sync with the line frequency. This then causes the DSL interface to restart and a flashing light should indicate this on the modem. Once it has retrained, the flashing light should remain solid and depending on how your ISP provisions your service will depend on what your modem does next. Usually, the modem will login to the ISP and obtain an IP address from the ISP.
If you have a noisy line, there could be interference causing an issue. You can often listen to this noise on ADSL lines by plugging in an analog telephone. Dial a single number so that the telco stops playing dial tone. You shouldnt be able to hear anything really but any hissing is noise. This is why the ISPs often reply to ADSL internet issues by asking "is the modem plugged into the master socket" as often the master socket in the home is extended internally and wiring gets knocked by vacuum cleaners etc or are just installed poorly.
The other bet I would hedge for $5 would be that the modem is not that great spec, has low CPU and memory resources or software bug and the whole device is either locking up or crashing, forcing a reboot. This will cause added down time because the modem takes time to restart and load the software and only after a completed boot will the DSL line go through the sync process and subsequently log into the internet line.
If you are fine to keep a wired connection, then it will help rule out wifi drops also. I suggest to check the link on the first reply because it (hopefully) will guide you through some easy troubleshooting steps that will usually confirm where the issue is.
Though if you want my best-guess then I would put $10 on the ADSL dropping during the download and having issues retraining to the DSL. Often, ISPs want to "show off" and give their customers the absolute fastest speed internet that they can at the cost of stability and reliability. My bet is that, during the download which slams the line, the modem is losing sync with the line frequency. This then causes the DSL interface to restart and a flashing light should indicate this on the modem. Once it has retrained, the flashing light should remain solid and depending on how your ISP provisions your service will depend on what your modem does next. Usually, the modem will login to the ISP and obtain an IP address from the ISP.
If you have a noisy line, there could be interference causing an issue. You can often listen to this noise on ADSL lines by plugging in an analog telephone. Dial a single number so that the telco stops playing dial tone. You shouldnt be able to hear anything really but any hissing is noise. This is why the ISPs often reply to ADSL internet issues by asking "is the modem plugged into the master socket" as often the master socket in the home is extended internally and wiring gets knocked by vacuum cleaners etc or are just installed poorly.
The other bet I would hedge for $5 would be that the modem is not that great spec, has low CPU and memory resources or software bug and the whole device is either locking up or crashing, forcing a reboot. This will cause added down time because the modem takes time to restart and load the software and only after a completed boot will the DSL line go through the sync process and subsequently log into the internet line.
If you are fine to keep a wired connection, then it will help rule out wifi drops also. I suggest to check the link on the first reply because it (hopefully) will guide you through some easy troubleshooting steps that will usually confirm where the issue is.

Thanks for posting back because this was new to me. Glad you got going