Bulk Capacitor for Full Wave Rectification
Let me give you an answer since I am the author of the note you gave as a reference:

What is the maximum limit for the bulk ripple: there is no real specification and a rule of thumbs given by Unitrode a while ago was 20-30% of the peak line voltage. Therefore, for a 85-V rms voltage (120 V peak roughly), you would size the capacitor value for a valley voltage of 85 dc for instance. I have seen many commercial designs in which the valley voltage was down to 60 V or so. The reason? A smaller bulk capacitor which drives size and cost down. Let's see a brief simulated example:

\$P_{out}=117\;W\$ , \$C_{bulk}=220\;µF\$ then \$I_{C,rms}=2.2\;A\$ and \$V_{valley}=86\;V\$
Now select a 100-µF capacitor:
\$P_{out}=117\;W\$ , \$C_{bulk}=100\;µF\$ then \$I_{C,rms}=1.9\;A\$ and \$V_{valley}=48\;V\$
So the rms current does not drastically change but the valley voltage does.

What is the impact of this large ripple: there are many issues brought by a low-value bulk capacitor:


the valley voltage sets the minimum operating voltage at which the power supply must deliver 100% of its rated power. It is often overlooked by designers but if you plan on designing a 100-W power supply unit (PSU) at a 85-V rms input with a 50-V valley voltage, then the PSU must be dimensioned for an operation at 50 V dc with some margin. So you realize the burden on the power supply (rms currents, conduction losses etc.) for a design at 50 V.
with such a low input voltage, the controller will push the duty ratio quite far and you may need a strong compensation ramp in a current-mode-controlled design. This extra ramp component may affect the maximum peak current and require to tweak the sense resistance to still pass power.
the worse-case right-half-plane-zero (RHPZ) will be very low, severely hampering the possible crossover frequency which cannot exceed 20-30% of the lowest RHPZ position. This is really an impeding factor.
you'll have to really increase the operating peak current in the bulk valley voltage, bringing extra burden on the transformer with saturation risks at high temperature. The real plague is the over-power phenomenon (OPP) that needs to be harnessed for the mandatory limited power-source test (LPS). By increasing the power capability at 85-V rms because of a high ripple voltage, you will pass significantly more power at 265 V rms, forcing you to find a way to fold the peak current limit in high-line conditions. Furthermore, the compensated OPP curve might approach a bell curve with a peak of power in the middle of the input voltage range if you over compensate the converter.
there is less energy storage with a small bulk capacitor and any small power drop will pass through and be observed on \$V_{out}\$.


What will happen if the voltage goes lower than 68 V in the example: well, either the PSU will go in protection mode because a) you've hit the maximum peak current limit too long or b) the power supply won't be able to regulate and a lot of ripple will be seen on \$V_{out}\$. If this lasts too long, temperature runaway may also happen with all consequences.

As a conclusion, do not neglect the selection of the bulk capacitor as it is one of the failing components in switching power supplies: too low a rms current capability, too hot, too close to a heatsink etc. Finally, select a serious brand for this capacitor.